Gromov Compactness Theorem for Pseudoholomorphic Curves

Treball ﬁnal de m`aster

MASTER` DE MATEMATICA` AVANC¸ADA

Facultat de Matem`atiques Universitat de Barcelona

Gromov compactness theorem for pseudoholomorphic curves

Autor: Carlos S´aezCalvo

Director: Dr. Ignasi Mundet i Riera Realitzat a: Departament de Algebra` i Geometria

Barcelona, 14 de setembre de 2014

Contents

Introduction 5

1 Introduction to symplectic and almost complex manifolds 7 1.1 Symplectic vector spaces ...... 7 1.2 Symplectic manifolds ...... 10 1.3 Complex manifolds and almost complex structures ...... 12 1.4 Kaehler manifolds ...... 20 1.5 Counterexamples ...... 22

2 Moduli spaces of pseudoholomorphic curves 25 2.1 Pseudoholomorphic curves ...... 25 2.2 Moduli spaces of pseudoholomorphic curves: a quick overview ...... 29 2.3 Failure of compactness ...... 32

3 The Gromov compactness theorem 35 3.1 Moduli space of stable curves ...... 35 3.2 Statement of the compactness theorem ...... 39

4 Local estimates 41 4.1 Review of Sobolev spaces and elliptic regularity ...... 41 4.2 Estimates on disks ...... 45 4.3 Estimates on cylinders ...... 50 4.4 Removal of singularities ...... 55

5 Proof of the Gromov compactness theorem 65 5.1 Bubbling ...... 65 5.2 Soft rescaling ...... 69 5.3 Gromov compactness ...... 76

6 The non-squeezing theorem 81 6.1 The theorem and its consequences ...... 81 6.2 Proof of the theorem ...... 83

Bibliography 89

3 4 CONTENTS Introduction

The main goal of this master thesis is to give a self-contained proof of the Gromov compactness theorem for pseudoholomorphic curves and the non-squeezing theorem in symplectic topology. Pseudoholomorphic curves are smooth maps from a Riemann surface into an almost complex manifold that respect the almost complex structures. If the target manifold is a complex manifold, we recover the notion of holomorphic maps, so pseudoholomorphic maps can be seen as the generalization of holomorphic maps to the almost complex setting. Pseudoholomorphic curves were introduced by Gromov in a ground-breaking paper published in 1985, [Gro]. Since then, they have become one of the main tools in the ﬁeld of symplectic topology.

Any symplectic manifold can be endowed with an almost complex structure that satisfies a condition of compatibility with the symplectic form. In this setting, one can consider moduli spaces of pseudoholomorphic curves, which in most situations are manifolds of finite dimension. However, in general these moduli spaces are not compact, and the problem of seeking a natural compactification for these moduli spaces arises. This problem was first solved by Gromov in his paper [Gro], and a complete picture of the compactified moduli space was finally given by Kontsevich in terms of stable maps. Essentially, the Gromov compactness theorem asserts that the failure of compactness of the moduli spaces of pseudoholomorphic curves can happen only in a very controlled way.

Compactness of the moduli spaces is crucial for applications in symplectic topology. One of the important results of Gromov in [Gro] was a proof of the non-squeezing theorem, asserting that a ball can be symplectically embedded in a symplectic cylinder if and only if the radius of the ball is smaller than the radius of the cylinder. Its importance lies in the fact that this leads to the deﬁnition of non-trivial global symplectic invariants, while it has been known for a long time that there are no local symplectic invariants. Moreover, compactness of the moduli spaces also play a crucial role for further developments of the theory, such as the deﬁnition of Gromov-Witten invariants or Floer homology.

The main novelty of this thesis, compared to the proofs of the compactness theorem that can be found in the literature, lies in the fact that we use exclusively the L2 theory of Sobolev spaces, while the usual proofs use either Lp theory of Sobolev spaces (for a general p) or isoperimetric inequalities.

The thesis is divided into several chapters. In the ﬁrst one we introduce all the elements of symplectic geometry necessary for the rest of the work. In particular, we introduce symplectic manifolds, complex manifolds, almost complex manifolds and Kaehler manifolds, studying its properties and the relations between them.

5 6 CONTENTS

In the second chapter, we introduce our main object of interest, pseudoholomorphic curves. Af- ter studying its basic properties, we organize them in a topological space, the moduli space of pseudoholomorphic curves. An important property of the moduli spaces of pseudoholomorphic curves is that, under suitable conditions, they are ﬁnite-dimensional smooth manifolds. However, this result needs some analytic machinery that is beyond the scope of this work. Therefore, we only give a brief idea of how this can be proved. In the last section, we see that these moduli spaces are not, in general, compact. Therefore, the need for a geometrically meaningful com- pactiﬁcation of the moduli spaces arises.

In the third chapter, we introduce the objects that form the compactiﬁcation of the moduli spaces, namely, the stable curves. We introduce the moduli spaces of stable curves and give them a suitable topology. In the last section of the chapter, we give the statement of the Gro- mov compactness theorem. For the notations and deﬁnitions for stable curves, we follow closely the reference [M-S1].

The fourth chapter constitutes the main contribution of this thesis. In this chapter of a technical nature, we develop estimates for pseudoholomorphic curves needed for the proof of the compactness theorem using only L2 methods. We use the obtained estimates to give a self-contained proof of the removal of singularities theorem, which is an analogue of the Riemann extension theorem for holomorphic functions (asserting that any bounded holomorphic function deﬁned in a punctured disk extends uniquely to a holomorphic function in the whole disk) in the pseudoholomorphic setting.

The ﬁfth chapter uses the results obtained in the previous chapter in order to give a complete proof of the Gromov compactness theorem. The proof also follows closely [M-S1].

In the last chapter, we give a proof of the Gromov’s non-squeezing theorem and discuss its importance. In particular, we use the theorem to deﬁne symplectic invariants. Our proof is essentially the same given by Gromov in [Gro], but with more detail. Our exposition follows closely that of [Hum].

Finally, I want to thank my advisor Ignasi Mundet i Riera for his guidance and ideas. Chapter 1

Introduction to symplectic and almost complex manifolds

This chapter is a short introduction to symplectic geometry, where we cover the basics and the topics we need for our exposition of pseudoholomorphic curves and the Gromov compactness theorem. In particular, we discuss symplectic manifolds, almost complex manifolds, complex manifolds and how all this concepts relate to each other. In the last section, we give a very brief overview of K¨ahlermanifolds, which are a big source of examples of symplectic manifolds.

1.1 Symplectic vector spaces

We start our exposition of symplectic geometry by studying the linear case, that is, a vector space with a non degenerate alternating bilinear form. We will see later that most of the results obtained in this linear setting extend to symplectic manifolds.

Deﬁnition 1.1.1. Let V be a ﬁnite dimensional vector space over R. A bilinear form ω satisfying the two following conditions:

1. (Alternating) ω(v, w) = −ω(w, v) for all v, w ∈ V

2. (Non degenerate) If ω(v, w) = 0 for all w ∈ V , then v = 0 is called a symplectic form, and the pair (V, ω) is called a symplectic vector space.

Observe that an alternating bilinear form in V is the same as a 2-form in V . We introduce now our main examples of symplectic vector spaces.

Example 1.1.2. Consider the vector space R2n (of dimension 2n), and write the standard basis as e1, ..., en, f1, ..., fn. We deﬁne

n X ω0 = ei ∧ fi i=1

Clearly ω is an alternating bilinear form, and it is easy to see that it is non degenerate. Indeed, observe that if v = a1e1 + ... + anen + b1f1 + ... + bnfn with an, bn ∈ R, then ω0(v, ei) = −bi

7 8CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

and ω0(v, fi) = ai. Therefore, if ω0(v, w) = 0 for all w ∈ V , then 0 = ω0(v, ei) = −bi and 0 = ω0(v, fi) = ai, so v = 0.

2n Therefore (R , ω0) is a symplectic vector space for each n.

Proposition 1.1.3. Let V be a vector space and let ω be a bilinear alternating form in V (maybe degenerate). Then, there exists a basis of V ,u1, ..., un, v1, ..., vn, w1, ..., wk, such that Pn ω = i=1 ui ∧ vi. Proof. Let us deﬁne ker(ω) = {v ∈ V : ω(v, ·) = 0}. Observe that ker(ω) is a subspace of V . Let w1, ..., wk be a basis for ker(ω). Choose now a complementary subspace to ker(ω), W , so that V = ker(ω) ⊕ W . Observe that ω|W is non degenerate. Take u1 ∈ W such that u1 6= 0. Since ω|W is non degenerate, there exists a vector v1 ∈ W such that ω(u1, v1) = 1. Deﬁne now ω W1 =< u1, v1 > and W1 = {v ∈ V : ω(v, w) = 0 for all w ∈ W1}.

ω ω We claim that W1 ∩ W1 = 0. Indeed, if v ∈ W1 ∩ W1 , we have v = au1 + bv1. But 0 = ω(v, u1) = aω(u1, u1) + bω(v1, u1) = −b and 0 = ω(v, v1) = aω(u1, v1) + bω(v1, v1) = a, so v = 0 as wanted.

ω Moreover, W = W1 ⊕ W1 . To see this, let v ∈ W and let a = ω(v, u1), b = ω(v, v1). Then, ω v = (−au1 + bv1) + (v + au1 − bv1). It is clear that −au1 + bv1 ∈ W1 and v + au1 − bv1 ∈ W1 by seeing that ω(v + au1 − bv1, u1) = ω(v + au1 − bv1, v1) = 0.

ω Observe now that ω| ω is non degenerate, because if v ∈ W satisfies ω(v, w) = 0 for all W1 1 ω ω w ∈ W1 , since ω(v, u1) = ω(v, v1) = 0 by the definition of W1 , we have that ω(v, w) = 0 for all w ∈ W1, so v = 0. Therefore, we can repeat the process and finish the proof by induction, since dim V < ∞.

For the last statement, observe that by construction, ω(ui, vj) = δij, ω(ui, uj) = ω(vi, vj) = 0 Pn and ω(ui, wj) = ω(vi, wj) = 0. Therefore, we have ω = i=1 ui ∧ vi. This proposition gives us immediately the following corollary.

Corollary 1.1.4. Let (V, ω) be a symplectic vector space. Then dim V is even. Moreover, this proposition tells us that any symplectic vector space is essentially one of the symplectic vector spaces from example 1.1.2. Deﬁnition 1.1.5. Let (V, ω), (V 0, ω0) be two symplectic vector spaces. A map φ : V −→ V 0 is called a linear symplectomorphism if it is a linear isomorphism that satisﬁes φ∗ω0 = ω (that is, ω0(φ(v), φ(w)) = ω(v, w) for all v, w ∈ V ). If there exists a linear symplectomorphism between two symplectic vector spaces, we say that the vector spaces are symplectomorphic.

Corollary 1.1.6. Let (V, ω) be a symplectic vector space of dimension 2n. Then, V is symplec- 2n tomorphic to (R , ω0).

Proof. Consider the basis u1, ..., un, v1, ..., vn of V given by the previous proposition (observe that since ω is non degenerate, we have k = 0). Deﬁne a linear symplectomorphism φ : V −→ R2n by 2n φ(ui) = ei, φ(vi) = fi. Since φ takes a basis of V to a basis of R , φ is an isomorphism. Since Pn Pn ∗ ω = i=1 ui ∧ vi and ω0 = i=1 ei ∧ fi, it is immediate to check that φ (ω0) = ω. 1.1. SYMPLECTIC VECTOR SPACES 9

As in the case of a vector space with an inner product, the symplectic form gives us a natural isomorphism between V and its dual. In fact, this is true for any vector space with a non degenerate bilinear form.

Proposition 1.1.7. Let V be a finite dimensional vector space and ω a non degenerate bilinear form in V . Then, the map ω˜ : V −→ V ∗ defined by ω˜(u)(v) = ω(u, v) is an isomorphism. Proof. ω˜ is linear because ω is bilinear. Suppose thatω ˜(u) = 0. Then, 0 =ω ˜(u)(v) = ω(u, v) for all v ∈ V . Therefore, since ω is non degenerate, we have u = 0. Soω ˜ is injective. Then, since V and V ∗ are vector spaces of the same dimension,ω ˜ is an isomorphism. In the symplectic setting, as in the inner product case, we can define the orthogonal of a subspace. Definition 1.1.8. Let (V, ω) be a symplectic vector space. If W is a subset of V , we define the symplectic complement of W as

W ω = {v ∈ V : ω(v, w) = 0 for all w ∈ W } However, the behaviour of the symplectic complement is very diﬀerent from the orthogonal in the inner product case. For instance, recall that if g is an inner product on a vector space V , we have that W ⊥ ∩ W = {0}.

2n Example 1.1.9. Consider (R , ω0).

ω ω 1. Let W =< e1, ..., ej > for some j ≤ n. Then, W =< e1, ..., en, fj+1, ..., fn > and W ⊂ W (W = W ω if j = n).

ω ω 2. Let W =< e1, ..., en, f1, ..., fj > for some j ≤ n. Then, W =< ej+1, ..., en > and W ⊂ W .

ω ω 3. Let W =< e1, f1 >. Then, W =< e2, ..., en, f2, ..., fn > and W ∩ W = {0}.

This example illustrates the most important types of subspaces in a symplectic vector space. Each of these types of subspaces receive a name. Deﬁnition 1.1.10. Let (V, ω) be a symplectic vector space, and let W ⊂ V be a subspace.

1. W is an isotropic subspace if W ⊂ W ω. 2. W is a coisotropic subspace if W ω ⊂ W . 3. W is a lagrangian subspace if W = W ω. 4. W is a symplectic subspace if W ∩ W ω = 0.

The previous example, together with the fact that any symplectic vector space is symplecto- 2n morphic to some (R , ω0), shows that any symplectic vector space has subspaces of all kinds described in the deﬁnition. Note also that there exist in general subspaces that are of none of the above types.

However, an important feature shared by both the orthogonal and the symplectic complement, is the following. 10CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

Proposition 1.1.11. Let (V, ω) be a symplectic vector space. Let W ⊂ V be a subspace. Then

dim W + dim W ω = dim V

∗ Proof. Consider the map Φ : V −→ W deﬁned by Φ(u) = ω(u, ·)|W . We have ker Φ = {u ∈ V : ω(u, v) = 0 for all v ∈ W } = W ω. Moreover, Φ is exhaustive, since we have seen in proposition 1.1.7 thatω ˜ is exhaustive. Since dim W ∗ = dim W , we have dim V = dim ker Φ + dim Φ(V ) = dim W ω + dim W .

dim V In particular, it follows that if W is a lagrangian subspace, then dim W = 2 , and conversely, dim V that a isotropic subspace is lagrangian if its dimension is 2 . To ﬁnish this section, we observe that any symplectic vector space comes equipped with a natural choice of orientation.

Proposition 1.1.12. Let V be a vector space, and let ω be an alternating bilinear form. Then ω is non degenerate if and only if ωn := ω∧ ...n) ∧ω 6= 0. In particular, if ω is a symplectic form, V has a natural choice of orientation.

Proof. Suppose ﬁrst that ω is degenerate. Then, there is some 0 6= v ∈ V such that ω(v, ·) = 0. n n Pick a basis v, u2, ..., uk of V . Then ω (v, u2, ..., uk) = 0, so ω = 0. Conversely, suppose ω is Pn non degenerate. Then, by 1.1.3, V has a basis u1, ..., un, v1, ..., vn such that ω = i=1 ui ∧ vi. n 1 Then, ω = n! u1 ∧ v1 ∧ ... ∧ un ∧ vn 6= 0. The last assertion follows from the fact that a choose non zero 2n-form is equivalent to a choose of orientation of V .

1.2 Symplectic manifolds

Deﬁnition 1.2.1. Let X be a smooth manifold, and let ω be a 2-form in X. ω is called a symplectic form in X if it is non degenerate (i.e., ωp is a non degenerate 2-form in TpX for every p ∈ X) and ω is closed (i.e., dω = 0).

A smooth manifold X endowed with a symplectic form ω is called a symplectic manifold.

From the work done in the previous section, we can draw immediately some facts about symplectic manifold.

Proposition 1.2.2. Let (X, ω) be a symplectic manifold. Then X has even dimension and it is an orientable manifold.

Proof. Since dim X = dim TpX for any p ∈ X, and (TpM, ωp) is a symplectic vector space, we have that dim X is even. Moreover, if dim X = 2n, ωn = ω∧ ...n) ∧ω is a 2n-form on X which is non zero at every point of X. Therefore ωn is a volume form, and X is orientable.

It follows from this proposition that not all manifolds admit a symplectic form. In the case of compact manifolds, the fact that ω is closed implies further topological constraints on X.

2 Proposition 1.2.3. Let (X, ω) be a compact symplectic manifold. Then, HdeR(X) 6= 0, and 2 0 6= [ω] ∈ HdeR(X) 1.2. SYMPLECTIC MANIFOLDS 11

Proof. Since ω is a closed 2-form, it represents a second de Rham cohomology class. It is enough to see that [ω] 6= 0, that is, that ω cannot be exact. Suppose to the contrary that ω is exact. n R n n Then, [ω ] = 0, so X ω = 0. But this is a contradiction with the fact that ω is a volume form in X. Therefore, if X is a compact manifold admitting a symplectic form, it has non zero second de Rham cohomology group. In particular, no Sn with n 6= 2 admits a symplectic structure. We see now that S2 does admit a symplectic form. In fact, much more is true:

Proposition 1.2.4. Let X be an orientable surface. Then, X admits a symplectic form. Proof. Since X is orientable, X has an area form dA, that is, a nowhere vanishing 2-form. We claim that ω = dA is a symplectic form. Indeed, dAp is non-degenerate by definition of area form, and it is closed because dω ∈ Ω3(X) = 0, since X is a surface. The main example of a non compact symplectic manifold is the following: 2n Example 1.2.5. Consider X = R with coordinates x1, ..., xn, y1, ..., yn. Define the 2-form Pn ω0 = i=1 dxi ∧ dyi. Then ω0 is clearly a symplectic form. We will call ω0 the standard symplectic form on R2n. We will now state a deep fact that completely determines the local structure of symplectic manifolds. Definition 1.2.6. Let (X, ω) and (X0, ω0) be symplectic manifolds. A map φ : X −→ X0 is a ∗ 0 symplectomorphism if it is a diffeomorphism and φ (ω ) = ω (that is, ωp(u, v) = ωφ(p)(dφp(u), dφp(v)) for all u, v ∈ TpX).

Theorem 1.2.7 (Darboux). Let (X, ω) be a symplectic manifold of dimension 2n. For any p ∈ X there exists a neighbourhood U of p such that (U, ω|U ) is symplectomorphic to an open subset of R2n with its standard symplectic form. In particular, there are local coordinates Pn x1, ..., xn, y1, ..., yn near p such that ω = i=1 dxi ∧ dyi.

The proof is not diﬃcult, using the formula of Cartan, LX = ιX d + dιX . For a proof, see any book on symplectic geometry, like [M-S2] or [Sil].

Observe that Darboux’s theorem tells us that all symplectic manifolds of the same dimension are locally the same, that is, there are no local symplectic invariants. This is a sharp contrast with the analogous case of Riemannian manifolds, since there we have the curvature as a local invariant. In consequence, all symplectic invariants must be global. Towards the end of this thesis, we will define a symplectic invariant, but in order to show that the definition makes sense, we will have to use the machinery of pseudoholomorphic curves, which are global objects defined in a symplectic manifold.

We finish this section by defining some important classes of submanifolds of a symplectic manifold. They correspond with the special types of subspace of a symplectic vector space defined in the previous section. Definition 1.2.8. Let (X, ω) be a symplectic manifold, and let Y ⊂ X be an embedded submanifold, with i : Y −→ X the inclusion map. Then Y is a lagrangian (resp. isotropic, coisotropic, symplectic) submanifold, if TpY ⊂ TpX is a lagrangian (resp. isotropic, coisotropic, symplectic) subspace, for each p ∈ Y . 12CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

1.3 Complex manifolds and almost complex structures

Deﬁnition 1.3.1. Let V be a ﬁnite dimensional vector space over R. A complex structure on V is an automorphism J : V −→ V such that J 2 = −Id.

A ﬁrst observation is that if V has a complex structure, then V must have even dimension. Indeed, if n = dim V , from J 2 = −Id we obtain (det J)2 = (−1)n. Since det J is a real number, this is possible only if n is even.

Observe also that if V is a real vector space with dim V = 2n and V has a complex structure, then it can be seen as a complex vector space of (complex) dimension n in the following way: (x + iy)v = xv + yJv. Hence the name of complex structure.

Conversely, if V is a real vector space of dimension 2n, then there exists a complex structure on V . Indeed, ﬁx a basis e1, ..., en, f1, ..., fn and consider the linear map J : V −→ V such that J(ej) = fj, J(fj) = −ej Then, it is immediate to check that J is an automorphism and J 2 = −Id.

We are mainly interested in vector spaces with a symplectic structure. In this setting, we would want a complex structure in V to be compatible in some sense with the symplectic form. In this respect, there are two relevant deﬁnitions.

Deﬁnition 1.3.2. Let (V, ω) be a symplectic vector space, and let J be a complex structure on V . We say that J is ω-tame if we have ω(v, Jv) > 0 for all v 6= 0 in V . We say that J is ω-compatible if it is ω-tame, and in addition ω(Jv, Jw) = ω(v, w) for all v, w ∈ V .

We have seen before that every vector space of even dimension admits a complex structure. Re- ﬁning our argument, it is easy to see that every symplectic vector space admits an ω-compatible complex structure. Just take as a basis for V that given by proposition 1.1.3 and deﬁne again J : V −→ V by J(ej) = fj, J(fj) = −ej. Then, it is immediate to check that J is ω-compatible.

However, we will present another construction that gives us lots of diﬀerent ω-compatible complex structures on J and that will be useful when we deal with almost complex structures on manifolds.

Proposition 1.3.3. Let (V, ω) be a symplectic vector space. Then, there exist ω-compatible complex structures J on V .

Proof. We start by choosing any inner product g on V . Then, observe that since g and ω are non degenerate, there is a linear map A : V −→ V such that ω(u, v) = g(Au, v) for all u, v ∈ V .

This map A satisﬁes the following properties:

1. A is skew-symmetric with respect to g: g(A∗u, v) = g(u, Av) = g(Av, u) = ω(v, u) = −ω(u, v) = g(−Au, v), so A∗ = −A.

2. AA∗ is symmetric: (AA∗)∗ = AA∗

3. AA∗ is positive: g(AA∗u, u) = g(A∗u, A∗u) > 0 if u 6= 0.

Since AA∗ is symmetric and positive, ir diagonalizes by the spectral theorem: ∗ −1 AA = B diag(λ1, ..., λ2n)B 1.3. COMPLEX MANIFOLDS AND ALMOST COMPLEX STRUCTURES 13

√ √ √ √ √ ∗ ∗ −1 ∗ Therefore, AA exists: AA = B diag( λ1, ..., λ2n)B . Moreover, observe that √AA ∗ does not depend on B nor on the√ order of the√ eigenvuales λi. Indeed, we can characterize AA ∗ as the linear map that acts as AA (u) = λiu if u ∈ Vλi where Vλi is the eigenspace of eigen- ∗ value λi of AA . √ √ Deﬁning J = ( AA∗)−1A, we obtain the polar decomposition of A: A = AA∗J.

∗ ∗ ∗ Since√A commutes with A (because A = −A), we have√ that AA commutes√ with A√. There- fore AA∗ commutes also with A: if u ∈ V then AA∗Au = λ AA∗u = λ λ u and √ √ √ λi i i i ∗ A AA u = λiAu = λiλiu. Hence, A commutes with J.

Finally, observe that J is orthogonal √ √ √ JJ ∗ = ( AA∗)−1AA∗( AA∗)−1 = ( AA∗)−2AA∗ = Id and skew-adjoint

√ √ √ J ∗ = J −1 = A−1 AA∗ = A−1( AA∗)−1AA∗ = ( AA∗)−1A∗ √ = −( AA∗)−1A = −J

With these properties, we can check that J is our wanted ω-compatible complex structure:

1. J 2 = −J(−J) = −JJ ∗ = −Id √ 2. ω(u, Ju) = g(Au, Ju) = g(−JAu, u) = g( AA∗u, u) > 0

3. ω(Ju, Jv) = g(AJu, Jv) = g(JAu, Jv) = g(Au, J ∗Jv) = g(Au, v) = ω(u, v)

This ﬁnishes the proof.

A very useful remark about the previous proof is that the almost complex structure J is obtained in a canonical way once we ﬁx the inner product g.

To ﬁnish with the linear theory, observe that if we have a symplectic vector space with a complex structure, then we get an inner product.

Proposition 1.3.4. Let (V, ω, J) be a symplectic vector space together with an ω-compatible complex structure. Then, g(u, v) = ω(Ju, v) is an inner product in V .

Proof. That g is a bilinear map is clear since ω is bilinear and J is linear. g is positive deﬁnite because g(u, u) = ω(Ju, u) > 0 by the ω-tameness. Finally, g is symmetric since g(u, v) = ω(Ju, v) = −ω(v, Ju) = −ω(−JJv, Ju) = ω(JJv, Ju) = ω(Jv, u) = g(v, u), where we have used the ω-compatibility.

We will extend now our results on complex structures to manifolds. The manifold analog of a complex structure is just a collection of complex structures in each tangent space which varies smoothly from point to point. 14CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

Deﬁnition 1.3.5. Let X be a smooth manifold. An almost complex structure in X is an automorphism of the tangent bundle of X, J : TX −→ TX such that J 2 = −Id.

A smooth manifold X together with an almost complex structure J is called an almost complex manifold.

Observe that then, an almost complex structure J in X induce a complex structure in each tangent space.

Let us study how almost complex structures behave under diﬀeomorphisms.

Definition 1.3.6. Let (X,J) be an almost complex manifold. Let Y be a smooth manifold, and −1 let ψ : X −→ Y a diffeomorphism. Then we define ψ∗(J) = dψ ◦ J ◦ (dψ) .

Since dψ is an isomorphism of tangent bundles, it is clear that ψ∗(J) is an almost complex structure in Y . Observe that we can use this to express an almost complex structure J on a manifold X in local coordinates: if (U, ξ) is a chart of X, then ξ∗(J|U ) is an almost complex structure on the open set ξ(U) ⊂ R2n.

As before, if (X, ω) is a symplectic manifold, then we are interested in the almost complex structures satisfying some kind of compatibility condition with the symplectic structure.

Deﬁnition 1.3.7. Let (X, ω) be a symplectic manifold, and let J be an almost complex structure on X. We say that J is ω-tame if we have ωp(v, Jv) > 0 for all v 6= 0 in TpX and for all p ∈ X. We say that J is ω-compatible if it is ω-tame, and in addition ω(Jv, Jw) = ωp(v, w) for all v, w ∈ TpX and all p ∈ X.

We will show now that any symplectic manifold X admits ω-compatible almost complex structures.

Proposition 1.3.8. Let (X, ω) be a symplectic manifold. Then, there exist ω-compatible almost complex structures on X.

Proof. Choose a Riemannian metric g on X. Now, we can apply at each tangent space TpX the construction from proposition 1.3.3 to the inner product gp in order to obtain an ωp-compatible complex structure Jp on TpX. Deﬁne J as J(p) = Jp. The only thing that is left is to check that J is smooth. But this follows from the fact that g is smooth and the construction from 1.3.3 is canonical. Following the proof of that result, we can therefore check that Jp depends smoothly on p.

As in the linear case, a symplectic manifold together with an ω-compatible almost complex structure determines a Riemannian metric on X.

Proposition 1.3.9. Let (X, ω, J) be a symplectic manifold together with an ω-compatible almost complex structure. Then, g deﬁned by gp(u, v) = ωp(Jpu, v) for u, v ∈ TpX is a Riemannian metric on X.

Proof. The proof is the same as in the linear case, except that here we have to check the smoothness of g. But this is immediate since both ω and J are smooth. 1.3. COMPLEX MANIFOLDS AND ALMOST COMPLEX STRUCTURES 15

We will denote the metric in this proposition by gJ .

We will now introduce the concept of a complex manifold and study its relations with almost complex structures. Deﬁnition 1.3.10. Let X be a topological manifold. A complex structure on X is a collection of n charts (Ui, ξi)i∈I , where Ui are open sets in X and ξi : Ui −→ V ⊂ C are homeomorphisms from n S Ui onto an open subset of C , such that i∈I Ui = X and for all i, j ∈ I such that Ui ∩ Uj 6= ∅ −1 we have that ξi ◦ ξj is a holomorphic map.

A manifold X endowed with such a complex structure is called a complex manifold of (complex) dimension n.

Observe that in particular, identifying Cn with R2n we have that a complex manifold of dimension n is a smooth manifold of dimension 2n. Indeed, under this identiﬁcation we have for a chart ξ(p) = (z1(p), ..., zn(p)) = (x1(p), y1(p), ..., xn(p), yn(p)), where zi : U −→ C are complex functions on the domain of the chart, and xi, yi : U −→ R are real functions. As in the real case, (z1, ..., zn) are called local coordinates near p if p is in the domain of the chart. The relation between zi and the pair xi, yi is given by zi = xi + iyi.

In the same way as smooth manifolds can be considered as the appropriate setting for doing calculus in the large, complex manifolds are the appropriate setting for doing complex analysis in the large. In particular, we can extend the notion of holomorphic map to maps between complex manifolds. Deﬁnition 1.3.11. Let X,Y be two complex manifolds. A map f : X −→ Y is said to be holomorphic at p ∈ X if there are charts (U, ξ) near p and (V, ψ) near f(p) such that ψ ◦ f ◦ ξ−1 is holomorphic at ξ(p). f is said to be holomorphic if it is holomorphic at every point of X. Observe that the previous deﬁnition is independent of the chosen charts because of the fact that the transition maps in complex manifolds are holomorphic.

In a complex manifold we can deﬁne several notions of tangent space at a point. Fix p ∈ X, and ﬁx local coordinates (z1, ..., zn) near p.

First of all, we can consider the tangent space of X at p as a 2n-dimensional smooth manifold, which we denote by TR,pX. This is a real vector space of dimension 2n with basis { ∂ , ∂ , ..., ∂ , ∂ }. ∂x1 ∂y1 ∂xn ∂yn

Secondly, we can consider the complexiﬁed tangent space TC,pX = TR,pX, which is a complex vector space of complex dimension 2n, with basis { ∂ , ∂ , ..., ∂ , ∂ }. ∂x1 ∂y1 ∂xn ∂yn Equivalently, we can consider the basis { ∂ , ∂ , ..., ∂ , ∂ }. ∂z1 ∂z1 ∂zn ∂zn

0 Finally, we can consider the holomorphic tangent space TpX, which is a complex vector space of complex dimension n, with basis { ∂ , ..., ∂ }. This is the natural tangent space to consider ∂z1 ∂zn when we consider X as a complex manifold.

A basic topological property of complex manifolds is that, like symplectic manifolds, they are always orientable. 16CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

Proposition 1.3.12. Let X be a complex manifold. Then X is orientable.

Proof. It suﬃces to check that if (U, ξ), (V, ξ0) are two complex charts, then the jacobian of the change ξ0 ◦ ξ−1 is always positive. Observe that, since ξ0 ◦ ξ−1 is a holomorphic function, it satisﬁes the Cauchy-Riemann equations. Using the basis ∂ , ..., ∂ , ∂ , ..., ∂ , writing ∂x1 ∂xn ∂y1 ∂yn 0 −1 ξ ◦ ξ = (ψ1, ..., ψn) = (u1, ..., un, v1, ..., vn), where ψi = ui + ivi, and using the Cauchy- Riemann equations ∂vi = ∂ui and ∂vi = − ∂ui to express everything in terms of x, we get that ∂yj ∂xj ∂xj ∂yj the jacobian of the change of variables is a sum of squares, hence positive.

The following proposition shows that every complex manifold carries a natural almost complex structure.

Proposition 1.3.13. Let X be a complex manifold. Then there exists a canonical choice of almost complex structure J on X.

Proof. Fix some chart U ⊂ X with local coordinates (z1, ..., zn) in X, and put zi = xi + iyi. Then, for p ∈ U we can consider the almost complex structure deﬁned by J(p)( ∂ ) = ∂ ∂xi ∂yi and I(p)( ∂ ) = − ∂ . We will check that J(p) is independent of the choice of local coordi- ∂yi ∂xi nates near p. Indeed, let (w1, ..., wn) be another set of local coordinates and put wi = ui + ivi. Let h = f + ig the change of coordinates from the wi’s to the zi’s. Recall that, since X is a complex manifold, h is holomorphic. Then we have that ∂ = P ∂fj ∂ + ∂gj ∂ and ∂ui j ∂ui ∂xj ∂ui ∂yj ∂ = P ∂fj ∂ + ∂gj ∂ . Since h is holomorphic, it satisﬁes the Cauchy-Riemann equations, ∂vi j ∂vi ∂xj ∂vi ∂yj hence ∂fj = ∂gj and ∂fj = − ∂gj . ∂ui ∂vi ∂vi ∂ui

So ﬁnally we have:

∂ X ∂fj ∂ ∂fj ∂ = − ∂u ∂u ∂x ∂v ∂y i j i j i j

∂ X ∂fj ∂ ∂fj ∂ = + ∂v ∂v ∂x ∂u ∂y i j i j i j

Therefore J(p)(( ∂ ) = P ∂fj J(p)( ∂ ) − ∂fj J(p)( ∂ ) = P ∂fj ∂ + ∂fj ∂ = ∂ and sim- ∂ui j ∂ui ∂xj ∂vi ∂yj j ∂ui ∂yj ∂vi ∂xj ∂vi ilarly J(p)( ∂ ) = − ∂ . ∂vi ∂ui

This shows that the deﬁnition of J is independent of the local coordinates, and since it is clearly smooth in each chart, this shows that J is an almost complex structure on X.

Note that the canonical almost structure of a complex manifold X acts as a rotation of π/2 in each plane { ∂ , ∂ }, or equivalently as the multiplication by i in this planes. Hence, the R ∂xi ∂yi name almost complex structure comes because it gives a notion of multiplication by i in a smooth manifold.

Now we can consider the inverse problem, that is, given an almost complex structure J on a smooth manifold X, there exists a complex structure on X such that J is the canonical almost complex structure induced by that complex structure? 1.3. COMPLEX MANIFOLDS AND ALMOST COMPLEX STRUCTURES 17

Definition 1.3.14. Let X be a smooth manifold and let J be an almost complex structure on X. J is said to be integrable if it is the canonical almost complex structure for some complex structure on X. In general, not all almost complex structures J are integrable. In fact, necessary and sufficient conditions for integrability were given by Newlander and Nirenberg. Definition 1.3.15. Given a map A : TX −→ TX, we define the Nijenhuis tensor as

2 NA(X,Y ) = −A [X,Y ] + A([AX, Y ] + [X, AY ]) − [AX, AY ]

One can easily check that NA is indeed a tensor.

Proposition 1.3.16 (Newlander-Nirenberg). An almost complex structure J : TX −→ TX is integrable if and only if NJ = 0. Therefore we see that integrability is equivalent to the fact that J satisﬁes a certain diﬀerential equation. The proof of this result can be found in theorem 11.8 of [Dem].

If we assume this result we can easily show that all orientable surfaces admit a complex structure.

Proposition 1.3.17. Let X be an orientable smooth surface. Then X admits a complex structure. Proof. First of all, observe that X admits an almost complex structure J. For instance, we have seen before that X admits a symplectic form, and that any symplectic manifold admits ω-compatible almost complex structures.

Now we only have to check that NJ = 0. Since N is a tensor, it is enough to check that NJ (a, b) = 0 for some basis a, b of TpX. Let 0 6= a ∈ TpX. Then {a, Ja} is a basis of TpX (otherwise, Ja = λa and ω(Ja, a) = 0, a contradiction with the ω-tameness of J). But then we have NJ (a, Ja) = [a, Ja] + J([Ja, Ja] + [a, JJa]) − [Ja, JJa] = [a, Ja] + J(−[a, a]) + [Ja, a] = [a, Ja] − [a, Ja] = 0. So by the Newlander-Nirenberg theorem, J is integrable. This shows that X admits a complex structure. We can now give several examples of complex manifolds. The most important ones are the manifolds of complex dimension 1. Deﬁnition 1.3.18. A connected complex manifold of dimension 1 is called a Riemann surface.

Example 1.3.19. Cn is a complex manifold of dimension n, with a complex structure given by the only chart (Cn, id). In particular, C is a Riemann surface. Example 1.3.20. Let Σ be a compact Riemann surface. Then, since Σ is an orientable smooth manifold of dimension 2, that is, a compact orientable smooth surface, we know by the classifi- cation theorem for surfaces that it is diffeomorphic to a sphere S2 or to a connected sum of tori, and therefore that topologically (and smoothly) they are classified according to their genus g. In fact, as we have seen in the previous proposition, one can construct Riemann surfaces of every genus, hence all compact orientable smooth surfaces admit a complex structure. The most important examples of compact complex manifolds are the complex projective spaces, which we now describe. 18CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

Example 1.3.21. Let CP n be the quotient space of Cn+1 − {0} by the equivalence relation (z0, z1, ..., zn) ∼ (w0, w1, .., wn) if and only if there exists a λ ∈ C, λ 6= 0, such that zi = λwi for all i = 1, ..., n. Alternatively, we can see it as the space of all lines in Cn+1. We can give n so-called homogeneous coordinates to CP by assigning to a point p = [(z0, ..., zn)] its coordinates [z0, ..., zn]. Note that then a point p has inﬁnitely many homogeneous coordinates, and [z0, ..., zn] and [w0, ..., wn] represent the same point if and only if zi = λwi for all i and some λ 6= 0.

As it is well-known from general topology, CP n is a Hausdorff compact topological space. We will see now that it has the structure of a complex manifold of dimension n. First of all, define n Ui = {p = [z0, ..., zn] ∈ CP : zi 6= 0}, and observe that it is well-defined (i.e., the condition zi 6= 0 is independent of the homogeneous coordinates that we choose for p).

n Sn n Clearly we have CP = i=0 Ui. We give maps ξi : Ui −→ C given by ξi([z0, ..., zn]) = z0 zi−1 zi+1 zn ( z , ..., z , z , ..., z ), which are seen to be homeomorphisms. Finally, the transition maps i i i i −1 z1 zi−1 1 zi+1 zn ξji := ξj◦ξ : ξi(Ui∩Uj) −→ ξj(Ui∩Uj) are given by ξji(z1, ..., zn) = , ..., , , , ..., , i zj zj zj zj zj where the i-th coordinate has been omited, which is a holomorphic map.

Therefore, CP n is a complex manifold of dimension n. Deﬁnition 1.3.22. Let (V, ω) be a symplectic vector space. We deﬁne J (V, ω) as the set of all ω-compatible complex structures on V , topologized as a subspace of End(V ).

Let (X, ω) be a symplectic manifold. We denote by J (ω) (or simply by J if ω is understood) the set of all the ω-compatible almost complex structures on X. Observe that an element of J is a section of the fibre bundle J −→ X with fibre over p, J (TpX, ωp). Therefore, we consider J (ω) as a topological space as the space of sections of the fibre bundle J −→ X. For the applications it will be very important the fact that J is path-connected. In fact, much more is true: J is contractible. There are several proofs of this fact. The one we give emphasizes the relation between lagrangian subspaces, almost complex structures and inner products.

Proposition 1.3.23. J (ω) is contractible. Proof. It is enough to check that the space J (V, ω) of ω-compatible complex structures on a symplectic vector space (V, ω), given the topology as a subspace of End(V ), is contractible. This is because then J (ω) is the space of sections of a ﬁbre bundle with contractible ﬁbers, hence it is also contractible.

Fix a lagrangian subspace L0 of (V, ω). Let L(V, ω, L0) be the space of all lagrangian subspaces of (V, ω) which intersect L0 transversally (that is, V = L0 + L). Let G(L0) be the space of all positive inner products on L0, and consider the map

Ψ: J (V, ω) −→ L(V, ω, L0) × G(L0) deﬁned by Ψ(J) = (JL0,GJ |L0 ).

First, note that Ψ is well-deﬁned. That is, JL0 ∈ L(V, ω, L0) and GJ |L0 ∈ G(L0). The second fact is immediate, since GJ is an inner product on V and the restriction of an inner product to any subspace is also an inner product. For the ﬁrst fact, we need to see that JL0 is a lagrangian ω subspace of V and that it intersects L0 transversally. Observe that if v ∈ L0 and w ∈ L0 = L0 ω we have ω(Jv, Jw) = ω(v, w) = 0, so (JL0) ⊂ JL0. Since J is an automorphism of V , we have 1.3. COMPLEX MANIFOLDS AND ALMOST COMPLEX STRUCTURES 19

dim JL0 = dim L0 = n/2, so JL0 is indeed a lagrangian subspace of V . Let us see that it intersects L0 transversally. Since dim L0 = dim JL0 = n/2 it is enough to see that L0∩JL0 = {0}. Let u ∈ L0∩JL0. Then, u = Jv for some v ∈ L0. So gJ (u, u) = ω(u, Ju) = ω(u, JJv) = −ω(u, v) = 0 ω because L0 = L0 . Since gJ is an inner product, it follows that u = 0.

Now, we see that Ψ is bijective. Let (L, G) ∈ L(V, ω, L0) × G(L0). We deﬁne J as follows. For ⊥ v ∈ L0, v = {u ∈ L0 : G(u, v) = 0} is an n − 1-dimensional subspace of L0. Its symplectic complement, (v⊥)ω is therefore (n + 1)-dimensional. Observe that (v⊥)ω ∩ L is 1-dimensional. ⊥ ω ω ⊥ Indeed, L ∩ L0 = {0} and L0 ⊂ (v ) (because L0 = L0 and v ⊂ L0), which together with the fact that (v⊥)ω is n+1-dimensional, implies that (v⊥)ω ∩L is 1-dimensional. Let Jv ∈ (v⊥)ω ∩L be the unique vector satisfying ω(v, Jv) = 1. Take now v1, ..., vn a G-orthonormal basis of L0, and let Jv1, ..., Jvn constructed as above. Note that Jv1, ..., Jvn are linearly independent, as follows from the fact that ω(vi, Jvj) = δij.

We claim that this deﬁnes the wanted J such that Ψ(J) = (L, G). Since Jvi ∈ L and L∩L0 = {0}, v1, ..., vn, Jv1, ..., Jvn is a basis of V and J is completely determined by deﬁning J(Jvi) = −vi. It is clear in this way that J is an automorphism of V satisfying J 2 = −1, that is, J is an almost complex structure on V . We have ω(vi, Jvi) = 1 and ω(Jvi, JJvi) = ω(Jvi, −vi) = ω(vi, Jvi) = 1, so ω J is ω-tame. Moreover, ω(Jvi, Jvj) = 0 = ω(vi, vj) (because Jvi, Jvj ∈ L = L ), ω(JJvi, Jvj) = −ω(vi, Jvj) = −δij = ω(Jvi, vj), and ω(JJvi, JJvj) = ω(vi, vj) = ω(Jvi, Jvj), so J is in fact ω-compatible.

Since L ∩ L0 = {0} and using the fact that v1, ..., vn is a basis of L0 and Jv1, ..., Jvn a basis of L, it follows that L = JL0. Finally, we must check that GJ |L0 = G. Indeed, G(vi, vj) = δij, while

GJ |L0 (vi, vj) = ω(vi, Jvj) = δij. So Ψ is exhaustive.

0 0 In order to see that Ψ is injective, suppose that Ψ(J) = Ψ(J ). Then, JL0 = J L0 =: 0 L and GJ |L0 = GJ |L0 =: G. Take v1, ...vn an orthonormal basis of L0 with respect to ⊥ G. Then, using the same notation as before, we have vi =< v1, ..., vi−1, vi+1, ..., vn > and ⊥ ω (vi ) =< v1, ..., vi−1, vi+1, ..., vn, ui > with ui ∈ L. Since Jvi ∈ L, ω(vi, Jvi) = 1 and G(vj, Jvi) = ω(vj, JJvi) = −ω(vj, vi) = 0 for all j 6= i, it follows that Jvi is the only vec- ⊥ ω 0 tor in (vi ) ∩ L satisfying ω(vi, Jvi) = 1, so J = J .

Observe that G(L0) is contractible (in fact, it is convex since tG1 + (1 − t)G2 is still an inner product on L0 for t ∈ [0, 1]). L(V, ω, L0) is also contractible. In order to see this, we observe that L(V, ω, L0) can be identiﬁed with the vector space of all symmetric n × n matrices, which is convex and hence contractible. Fix an ω-compatible almost complex structure J. Observe that if L is an n-dimensional subspace of V transversal to L0, we have that L is the graph of a linear map S : JL0 −→ L0 where L0 =< v1, ..., vn > and L =< Jv1 + SJv1, ..., Jvn + SJvn >. Observe that in these basis, the linear map S is expressed as a symmetric n × n matrix A. Indeed, we have Aij = ω(SJvi, Jvj). Then, using that L0 and L are Lagrangian, we have that 0 = ω(Jvi + SJvi, Jvj + SJvj) = ω(Jvi, Jvj) + ω(Jvi, SJvj) + ω(SJvi, Jvj) + ω(SJvi + SJvj) = −ω(SJvj, Jvi) + ω(SJvi, Jvj), so Aij = ω(SJvi, Jvj) = ω(SJvj, vi) = Aji and A is symmetric. Conversely, we can see similarly that if A is symmetric, then L is Lagrangian.

To ﬁnish, just observe that since Ψ is a bijection and we have just seen that L(V, ω, L0) × G(L0) is contractible, we have that J (V, ω) is contractible, as wanted. 20CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

1.4 Kaehler manifolds

The most important class of symplectic manifolds is the class of Kählermanifolds. In this section we will define them and give some examples of Kähler manifolds and of non-Kählersymplectic manifolds.

Definition 1.4.1. A symplectic manifold (X, ω, J) with an ω-compatible almost complex structure J is said to be a Kählermanifold if J is integrable. In this situation, the symplectic form ω is called a Kählerform.

There is an equivalent deﬁnition coming from complex geometry as a complex manifold equipped with an Hermitian metric.

Deﬁnition 1.4.2. Let X be a complex manifold. An Hermitian metric h in X is a map h : TX × TX −→ TX such that hp is a nondegenerate Hermitian form in TpX, that is, if P i j we write in local coordinates h = i,j hijdz ⊗ dz , hij is a positive deﬁnite Hermitian matrix.

A complex manifold X endowed with an Hermitian metric is called a Hermitian manifold.

Observe that in an Hermitian manifold, we have in a natural way a 2-form and a Riemannian i metric. Indeed, the Hermitian form is a 2-form defined as ω := 2 (h − h), and we have the 1 Riemannian metric g = 2 (h+h). In this way, we can express the Hermitian metric as h = g −iω. Definition 1.4.3. A complex manifold X together with a Hermitian metric H is called a Kähler manifold if its Hermitian form ω is closed.

Proposition 1.4.4. The two deﬁnitions of K¨ahlermanifold coincide.

Proof. (Sketch). We only indicate the relation between the K¨ahlerform and the hermitian metric.

It can be shown that if h is an Hermitian metric on X its Hermitian form ω is symplectic, and if J is the canonical almost complex structure, J is ω-compatible.

Conversely, if J is an integrable almost complex structure in X and ω is a symplectic form such that J is ω-compatible, then it is can be checked that h = gJ + iω is an Hermitian metric, and obviously it has a closed Hermitian form.

A very usual way to give a K¨ahlerstructure on a manifold is by means of a K¨ahlerpotential. Let us describe it.

Deﬁnition 1.4.5. Let X be a complex manifold. A function ρ : X −→ R is strictly plurisubhar- ∂2ρ monic (spsh) if, on each complex chart U with local coordinates z1, ..., zn, the matrix ∂z ∂z (p) i j ij is positive deﬁnite for all p ∈ U.

Proposition 1.4.6. Let X be a complex manifold and ρ : X −→ R a spsh function. Then i ω = ∂∂ρ 2 is a K¨ahlerform. 1.4. KAEHLER MANIFOLDS 21

Such a function ρ is then called a K¨ahlerpotential.

Locally, we also have the converse. That is, in a Kählermanifold, the Kählerform always derive locally from a Kählerpotential.

Theorem 1.4.7. Let ω be a K¨ahlerform of a K¨ahlermanifold X and let p ∈ X. Then there exist a neighbourhood U of p such that, on U,

i ω = ∂∂ρ 2 With this fact, we can prove that complex submanifolds of a K¨ahlermanifold are also K¨ahler manifolds.

Proposition 1.4.8. Let X be a complex manifold, ρ : X −→ R spsh and M a complex submanifold of X (that is, the inclusion i : M −→ X is a holomorphic embedding). Then i∗ρ is spsh.

Proof. Take a chart (U, z1, ..., zn) centered at p and adapted to M, that is, such that U ∩ M is ∗ given by z1 = ... = zm = 0 (where m = dim M. Then, we have that i ρ is spsh if and only if the ∂2ρ matrix (0, ..., 0, zm+1, ..., zn)i=m+1,...,n;j=m+1,...,n is positive definite, but this is a minor of ∂zi∂zj ∂2ρ the matrix (p))ij, which is positive definite by hypothesis, so it is also positive definite. ∂zi∂zj

Corollary 1.4.9. Any complex submanifold of a K¨ahlermanifold is also K¨ahler.

Now we can give lots of examples of K¨ahler(and hence, symplectic) manifolds.

Example 1.4.10. Consider in Cn the function ρ(z) = |z|2 = zz. This function is real valued ∂2ρ n and it is spsh, since (z)ij = δij. Therefore, it is a Kählerpotential and is a Kähler ∂zi∂zj C manifold. By the last corollary, all complex submanifolds of Cn are also Kählermanifolds.

Example 1.4.11. Let us return to CP n. We have seen that it is a complex manifold. We claim that it is also a Kählermanifold. Indeed, we can define a Kählerpotential in n by n n C 2 C i ∗ C ρFS := log(|z| + 1). Then, putting ωFS = 2 ∂∂ρ, we can write ωFS|Ui = φi (ωFS) where n φi : Ui −→ is the usual chart map. One can check that ωFS is then a well-defined Kähler C n n n ∗ C ∗ C n form in CP (that is, φi (ωFS) = φj (ωFS), called the Fubini-Study Kählerform. Therefore, CP is a Kählermanifold, as are all its complex submanifolds.

Example 1.4.12. [Smooth projective varieties] All smooth projective algebraic varieties (the locus of zeros of a set of homogeneous polynomials in CP n) are complex submanifolds of CP n (as can be easily seen by an application of the holomorphic inverse function theorem), and hence, a K¨ahlermanifold. In particular, smooth projective curves are K¨ahler manifolds. Curves have complex dimension 1, and hence they are Riemann surfaces. Since we already know that Riemann surfaces are symplectic manifolds, this gives another proof of the fact that smooth projective curves are symplectic manifolds. 22CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS

Kählermanifolds have been studied intensively, and there are plenty of interesting results about them. Here we will only quote one result, restricting the class of compact manifolds that admit a Kählerstructure, which can be proved by means of Hodge theory. Recall that the Betti numbers of X are defined as

k k b (X) := dim HdeR(X)

Theorem 1.4.13. If X is a compact K¨ahlermanifold, then the odd Betti numbers b2k+1 are even.

1.5 Counterexamples

In this chapter we have introduced several structures on manifolds: we have seen symplectic, almost complex, complex and K¨ahlermanifolds. We have pointed out several relations between them, that we summarize in the following proposition

Proposition 1.5.1. We have the following

(1) Any K¨ahlermanifold is a complex manifold.

(2) Any K¨ahlermanifold is a symplectic manifold.

(3) Any complex manifold is an almost complex manifold.

(4) Any symplectic manifold can be given the structure of an almost complex manifold.

We will give now examples (without proof that they satisfy the desired properties) showing that neither of the above implications can be reversed.

The following example is due to Thurston.

Example 1.5.2. [A symplectic and complex manifold that is not K¨ahler] Consider R4 with the usual symplectic form ω0 = dx1 ∧ dy1 + dx2 ∧ dy2. Let Γ be the discrete group of symplectomorphisms generated by the following symplectomorphisms:

γ1(x1, x2, y1, y2) = (x1, x2 + 1, y1, y2)

γ1(x1, x2, y1, y2) = (x1, x2, y1, y2 + 1)

γ1(x1, x2, y1, y2) = (x1 + 1, x2, y1, y2)

γ1(x1, x2, y1, y2) = (x1, x2 + y2, y1 + 1, y2) Consider now the compact manifold X = R4/Γ. It is symplectic with the symplectic form induced from ω0, and it can be shown that it admits a complex structure. However, it cannot 1 be K¨ahler, since π1(X) = Γ, so H (X; Z) = Γ/[Γ, Γ] which has rank 3. Therefore, b1 = 3, which implies that it cannot be K¨ahleras seen in theorem 1.4.13. 1.5. COUNTEREXAMPLES 23

The following example is due to Hopf. Example 1.5.3. [A complex manifold that does not admit any symplectic structure] Consider the compact manifold S1×S3, called the Hopf surface. It is not symplectic, since H2(S1×S3) = 0. However, it is complex, since S1 × S3 =∼ (C2 − {0})/Γ, where Γ = {2nid : n ∈ Z} is a discrete group of holomorphic transformations.

Example 1.5.4. [There are almost complex manifolds that are neither complex nor symplectic] Consider CP 2]CP 2]CP 2. It can be shown that this manifold admits an almost complex structure, but it is neither complex nor symplectic (proved by Taubes). There are also examples of symplectic manifold that does not admit any complex structure. Fern´andez-Gotay-Gray proved the existence of such manifolds. Their examples are circle bundles over circle bundles over a 2-torus. However we will not discuss them further here. 24CHAPTER 1. INTRODUCTION TO SYMPLECTIC AND ALMOST COMPLEX MANIFOLDS Chapter 2

Moduli spaces of pseudoholomorphic curves

In this chapter we introduce the main object of this thesis: pseudoholomorphic curve. After defining them, we prove some of its basic properties. In the second section, we show how they form moduli spaces and explain briefly how these moduli spaces have the structure of a finite dimensional manifold, for most situations. In the last section, we show by means of an example that these moduli spaces fail to be compact in general, and we will see how this failure of compactness occurs, introducing the phenomenon of bubbling.

2.1 Pseudoholomorphic curves

In all this section, let (X, ω) be a symplectic manifold, and let J be an ω-compatible almost complex structure on X. Moreover, denote by Σ a Riemann surface, by j the canonical almost complex structure on Σ and by [Σ] ∈ H2(Σ) the canonical choice of orientation induced by the canonical orientation of C via any holomorphic chart (since change of coordinates between holomorphic charts are orientation preserving, this orientation is well-defined). We also equip Σ with a volume form dvolΣ. Then, the Riemann surface Σ carries a natural metric determined ∂ by j and dvolΣ as follows. Pick local coordinates z = s + it near a point p ∈ Σ. Then, ∂s and ∂ ∂ ∂t = j ∂s form a local frame of T Σ. By rescaling the coordinates (using λz instead of z for λ ∈ C − {0}), we may assume that dvolΣ = ds ∧ dt. We then define a metric g on Σ by specifying ∂ ∂ that ∂s and ∂t form an orthonormal local frame. This gives as a well-defined metric, since using the fact that the change of coordinates is holomorphic one can check that the definition is independent of the chosen coordinates. Definition 2.1.1. A smooth map u :Σ −→ X is a J-holomorphic curve (or pseudoholomorphic curve if J is understood) if it satisfies du ◦ j = J ◦ du. Observe that du ◦ j = J ◦ du is equivalent to

du + J ◦ du ◦ j = 0 as is easily seen composing with J and using J 2 = −id.

In order to handle pseudoholomorphic curves it will be very useful to introduce the following two operators, adapted at the almost complex structure J.

25 26 CHAPTER 2. MODULI SPACES OF PSEUDOHOLOMORPHIC CURVES

¯ Deﬁnition 2.1.2. We deﬁne the operators ∂J and ∂J by: 1 ∂ u = (du − J ◦ du ◦ j) J 2 1 ∂¯ u = (du + J ◦ du ◦ j) J 2 ¯ Therefore, the pseudoholomorphicity condition can be expressed as ∂J u = 0. Observe also that ¯ for each u, ∂J u and ∂J u are 1-forms.

Now we are going to express the pseudoholomorphicity condition as a diﬀerential equation in terms of local coordinates. In order to get a good expression, we will use only holomorphic charts for Σ, that is, charts in the complex structure of Σ. Recall that if z = s + it are such local ∂ ∂ ∂ ∂ coordinates, we have that j( ∂s ) = ∂t and j( ∂t ) = − ∂s . Such local coordinates are also called conformal.

Proposition 2.1.3. Let (U, φα) be a chart in Σ with z = s + it as conformal local coordinates and put uα = u ◦ φα. In this coordinates, the condition du ◦ j = J ◦ du can be expressed as

∂suα + J(uα)∂tuα = 0 where as usual we still call J the expression of the almost complex structure on X in a given chart.

∂ ∂ Proof. Using the local coordinates, and recalling that in these coordinates j( ∂s ) = ∂t and ∂ ∂ j( ∂t ) = − ∂s we have: ∂ ∂ du + Jdu j = ∂ u + (J ◦ u )∂ u ∂s ∂s s α α t α ∂ ∂ du + Jdu j = ∂ u − (J ◦ u )∂ u ∂t ∂t t α α s α Therefore, in such a chart, the condition to be pseudoholomorphic reduces to the following two diﬀerential equations:

∂suα + (J ◦ uα)∂tuα = 0

∂tuα − (J ◦ uα)∂suα = 0 Now just observe that they are in fact the same equation, as seen composing one of them with J ◦ Uα, so the result follows. With this, we can see where the name pseudoholomorphic comes from. Assume that X is a complex manifold and J is its canonical almost complex structure. In this situation, choosing conformal coordinates for X also, so that J = J0 in these coordinates (where J0 is the canonical almost complex structure on Cn), the condition of pseudoholomorphicity in local coordinates reduce to the usual Cauchy-Riemann equations, and thus say that the map is holomorphic.

The above proof also shows 2.1. PSEUDOHOLOMORPHIC CURVES 27

Proposition 2.1.4. Let (U, φα) be a chart in Σ with z = s + it as conformal local coordinates and put uα = u ◦ φα. In this coordinates 1 1 ∂¯ u = (∂ u + J(u )∂ u )du + (∂ u − J(u )∂ u )dt J 2 s α α t α 2 t α α s α In general, the equations of the previous proposition are called non linear Cauchy-Riemann equations. They are non linear but elliptic, since its linearization are clearly the usual Cauchy- Riemann equations, which are well-known to be elliptic.

An essential property of a pseudoholomorphic curve is its energy, which we now deﬁne and give some properties. Deﬁnition 2.1.5. Let u :Σ −→ X be a smooth map. Its energy is Z 1 2 E(u) = |du|J dvolΣ 2 Σ where |du|J means the norm of du with respect to the metric gJ in X. 1 R 2 We will also use the notation E(u; A) := 2 A |du|J dvolΣ, for A ⊂ Σ. We will call energy density to |du|J .

Observe then that, by deﬁnition, the energy is just one half of the square of the L2 norm of du with respect to the metric gJ .

A trivial remark that we will use several times is that u is constant if and only if E(u) = 0.

We will see now that the energy of a pseudoholomorphic curve is a topological invariant if J is ω-compatible.

Deﬁnition 2.1.6. Let u :Σ −→ X be a pseudoholomorphic curve, and let [A] ∈ H2(X) be a homology class. We say that u represents [A] if u∗([Σ]) = [A], where u∗ : H2(Σ) −→ H2(X) is the induced map in the homology. The main proposition about the energy is the following identity.

Proposition 2.1.7. Let u :Σ −→ X be a smooth map, and J an ω-compatible almost complex structure on the symplectic manifold (X, ω). Then Z Z 2 ∗ E(u) = |∂J u|J dvolΣ + u ω Σ Σ where u∗ω represents the 2-form on Σ that is the pullback by the map u of the symplectic form ω. Proof. We choose conformal coordinates z = s + it on Σ, and assume without loss of generality that Σ is an open subset of C. We have, in this case

1 1 |du|2 dvol = (|∂ u|2 + |∂ u|2 )ds ∧ dt 2 J Σ 2 s J t J 1 = |∂ u + J∂ u|2 ds ∧ dt− < ∂ u, J∂ u > ds ∧ dt 2 s t J s t J 2 = |∂J u|J dvolΣ− < ∂su, J∂tu >J ds ∧ dt 28 CHAPTER 2. MODULI SPACES OF PSEUDOHOLOMORPHIC CURVES where we have used in the third equality the ω-compatibility of J.

∗ Observe that we have < ∂su, J∂tu >J = −ω(∂su, ∂tu) = −u ω.

Therefore, we ﬁnally get the formula from the statement.

Z Z 2 ∗ E(u) = |∂J u|J dvolΣ + u ω Σ Σ

Corollary 2.1.8. If u :Σ −→ X is a pseudoholomorphic curve, its energy is

Z E(u) = u∗ω Σ In particular, the energy of a pseudoholomorphic curve depends only on the homology class it represents.

Proof. The ﬁrst part follows immediately from the proposition, since being pseudoholomorphic implies that ∂J u = 0.

For the second part, observe that, since ω is closed it deﬁnes a cohomology class [ω]. Then, we have

Z E(u) = u∗ω =< u∗[ω], [Σ] >=< [ω], [A] > Σ where [A] is the homology class represented by u, and < −, − > represents the Kronecker pairing.

Observe also that it follows immediately from the formula in proposition 2.1.7 the following very important fact about pseudoholomorphic curves.

Corollary 2.1.9. Let u be a pseudoholomorphic map representing the homology class A. Then, the energy of u is minimal with respect to the energies of all smooth maps representing A.

Another very important property of the energy of a pseudoholomorphic curve is that it is conformal invariant, that is, it only depends on the almost complex structure j of Σ, but not on 2 the speciﬁc metric on Σ, determined by dvolΣ. However, the energy density |du|J does depend on the metric of Σ in general. This is indeed immediate from what we have done, since in fact we have seen that the energy of a pseudoholomorphic map only depends on topological data. Therefore, if we reparametrize a pseudoholomorphic map u :Σ −→ X by composing it with an automorphism of Σ, ψ ∈ Aut(Σ), we get E(u) = E(u ◦ ψ). This will be used extensively in the proof of the Gromov compactness theorem. 2.2. MODULI SPACES OF PSEUDOHOLOMORPHIC CURVES: A QUICK OVERVIEW 29

2.2 Moduli spaces of pseudoholomorphic curves: a quick overview

In this section we will describe the topology of the moduli spaces of pseudoholomorphic curves. Since the construction of moduli spaces and the proof of its properties (such as the fact that they are ﬁnite-dimensional manifolds) require advanced analytic tools such as Fredholm theory on Banach manifods, which are beyond the scope of this work, we will content ourselves to give a brief overview of the theory and the construction of moduli spaces. This overview will be enough for our purposes, and its speciﬁc details will not be very important for what follows, with the exception of the proof of the non-squeezing theorem, where we will make non-trivial use of some of the theorems stated here. For simplicity, we restrict ourselves to the genus 0 case.

We begin with a quick overview of Fredholm theory. First of all let us introduce the notion of Banach manifolds, which are essentially manifolds modelled on a Banach space instead of Rn. In particular, this allows us to speak about infinite-dimensional manifolds. Before introducing it, we recall the definition of differentiability of a map between Banach spaces.

Deﬁnition 2.2.1. Let V,W be Banach spaces. Let Ω ⊂ V be an open set, and let f :Ω −→ W be a map between them. We say that F is diﬀerentiable at a point x ∈ V if there exists a linear map Tx : V −→ W such that ||f(x + h) − f(x) − T h|| lim x = 0 ||h||→0 ||h||

That is, if the map Tx is a good linear aproximation near x. In this situation, Tx is called the diﬀerential of f at x.

We say that f is differentiable if it is differentiable at each x ∈ V . In this situation we say that T : V −→ W defined as T (x) := Tx is its differential. If moreover T is continuous, we say that f is of class C1. Similarly, if f is differentiable k times with continuity, we say that f is of class Ck (where k = ∞ means that f can be differentiated any number of times).

Deﬁnition 2.2.2. Let E be a Banach space. A Ck E-atlas on a topological space X consists on charts (Ui, ξi) such that the Ui form an open cover of X, ξi : Ui −→ ξi(Ui) is a homeomorphism −1 k from Ui onto an open subset ξi(Ui) ⊂ E such that ξj ◦ ξi : ξi(Ui ∩ Uj) −→ ξj(Ui ∩ Uj) is a C function.

Deﬁnition 2.2.3. A Banach Ck-manifold X is a Hausdorﬀ space having a Ck E-atlas, for some Banach space E.

A very important class of operators between Banach spaces is that of the Fredholm operators.

Deﬁnition 2.2.4. Let E,V be Banach spaces and F : X −→ Y a continuous operator between them. We say that F is a Fredholm map if both kerF and cokerF are ﬁnite-dimensional. In this case, we say that the index of F is indexF := dimkerF + dimcokerF .

As in the case of ﬁnite-dimensional manifolds, we can deﬁne the tangent space at a point of a Banach manifold.

Deﬁnition 2.2.5. Let X be a Banach manifold of class Ck, with k ≥ 1, and let p ∈ X. Con- sider triples (U, ϕ, v) where (U, ϕ) is a chart centred at p and v an element of the model Banach space. We deﬁne an equivalence relation on the set of all these triples specifying that (U, ϕ, v) 30 CHAPTER 2. MODULI SPACES OF PSEUDOHOLOMORPHIC CURVES and (U 0, ϕ0, v0) are equivalent if the derivative of ϕ0 ◦ ϕ−1 at ϕ(p) sends v to v0.

We define the tangent space TpX of X at p as the set of all such equivalence classes. Each chart (U, ϕ) centred at p gives a bijection of TpX with a Banach space, via (U, ϕ, v) is sent to v. This bijection gives TpX the structure of a Banach space. Likewise, we have a definition for the differential of a map between Banach manifolds. Definition 2.2.6. Let f : X −→ Y be a map between Banach manifolds. Then, the differential dfp : TpX −→ Tf(p)Y is the unique linear map satisfying that for (U, ϕ) a chart centred at p ∈ X and (V, ψ) a chart centred at f(p) ∈ Y with f(U) ⊂ V and v˜ ∈ TpX is represented in the chart −1 (U, ϕ) by v, then dfp(˜v) is represented in the chart (V, ψ) by the vector D(ψ ◦ f ◦ ϕ )(ϕ(p))v. We can extend the notion of a Fredholm operator to maps between Banach manifolds. This will be the main maps we will be interested in, since they are the maps for which most of the theorems from finite-dimensional differential topology extend to the infinite-dimensional setting. Definition 2.2.7. A C1-map φ : X −→ Y between Banach manifolds is called a Fredholm map if at each point p ∈ X, its differential dpφ : TpX −→ TpY is a Fredholm operator. We now consider analogues of finite dimensional theorems that hold for Banach manifolds and Fredholm maps between them. Recall that a subset of a topological space is called residual if it contains a countable intersection of open dense sets. In the case of a complete metric space we have that any residual set is dense, by the Baire category theorem. Definition 2.2.8. Let F : M −→ N be a map between Banach manifolds. We say that y ∈ N −1 is a regular value of F if dFx : TxM −→ TyN is surjective for all x ∈ F (y). Observe that if F −1(y) = ∅, y is a regular value of F .

Theorem 2.2.9 (Sard-Smale). The set of regular values of a C∞ Fredholm map F : M −→ N is residual in N.

Theorem 2.2.10 (Transversality theorem). Let F : M −→ N be a smooth Fredholm map and G : Y −→ N be an embedding of a finite-dimensional manifold Y into N. Suppose that F is transverse to G on a closed subset of W ⊂ Y . Then, there exists a map G0 : Y −→ N arbitrarily 1 0 0 close to G in the C -topology such that G is transverse to F and G |W = G. We are now ready to introduce the moduli spaces of pseudoholomorphic curves, give them a topology, and give a hint at the proof that they are finite-dimensional manifolds. Definition 2.2.11. Let (X, ω) be a compact manifold, J an ω-compatible almost complex structure on X and A ∈ H2(X) a second homology class of X. We define

2 2 M(A, J) = {u : S −→ X : u∗[S ] = A, df ◦ j = J ◦ df} That is, M(A, J) is the set of all J-holomorphic curves that represent A.

In order to have good properties on the moduli spaces (that is, in order to prove that they are ﬁnite-dimensional manifolds), we need to restrict ourselves to a special class of pseudoholomorphic curves, that we now introduce. 2.2. MODULI SPACES OF PSEUDOHOLOMORPHIC CURVES: A QUICK OVERVIEW 31

Deﬁnition 2.2.12. Let u :Σ −→ X be a J-holomorphic curve. We say that u is multiply covered if there exists a Riemann surface Σ0 and a holomorphic branched covering φ :Σ −→ Σ0 such that u = u0 ◦ φ, with deg(φ) > 1. If a pseudoholomorphic curve is not multiply covered, it is said to be simple.

Deﬁnition 2.2.13. Let (X, ω) be a compact manifold, J an ω-compatible almost complex structure on X and A ∈ H2(X) a second homology class of X. We deﬁne

∗ 2 2 M (A, J) = {u : S −→ X : u∗[S ] = A, df ◦ j = J ◦ df, u is simple} That is, M∗(A, J) is the set of all simple J-holomorphic curves that represent A.

Let us give a topology to the moduli spaces. Observe first that we can assume M(A, I) ⊂ C∞(S2,X). Now, C∞(S2,X) has a natural topology as a Fréchet space. Namely, we say that a α α 2 sequence fn converge to f if for all α D fn → D f uniformly on S . In this case we say that fn converges to f in the C∞ topology. Then, we define the C∞ topology on S2 by saying that a set A ⊂ C∞(S2,X) is closed if every convergent sequence in A converges to a function in A. One can check that this defines indeed a Fréchet topology on C∞(S2,X). Since M(A, J) ⊂ C∞(S2,X), M(A, J) is a topological space with the C∞ topology inherited from C∞(S2,X). Observe however, that C∞(S2,X) is not a Banach manifold.

Using the above techniques, it can be seen that M∗(A, J) with this topology is in fact a ﬁnite dimensional manifold for a dense set of J ∈ J . Such J’s are called regular, and we denote the ∗ space of all regular almost complex structures by Jreg. This is done by expressing M (A, J) as ¯−1 ¯ ∂J (0) (where we interpret ∂ as a Fredholm map between certain Banach manifolds, and therefore we have to consider an embedding of M∗(A, J) in a more general space than C∞(S2,X)), ¯ ∗ and using that for J ∈ Jreg 0 is a regular value of ∂J , hence M (A, J) is a ﬁnite dimensional manifold for such J’s.

For the applications, where we will need to consider several almost complex structures, it will be useful to introduce also the universal moduli space.

Deﬁnition 2.2.14. Let A ∈ H2(X) a homology class. We deﬁne the universal moduli space of pseudoholomorphic curves representing A as

∞ 2 ¯ 2 M(A, J ) := {(u, J) ∈ C (S ,X) × J : ∂J (u) = 0, u∗([S ]) = A} We denote by P : M(A, J ) −→ J the natural projection.

Observe that P −1(J) = M(A, J) × {J}, which is essentially the moduli space of J-holomorphic curves.

It turns out that M(A, J ) is a Banach manifold, the projection P is a Fredholm map and the set of regular values of P is Jreg.

Usually, however, we will not be interested just in pseudoholomorphic curves, but in unparametrized pseudoholomorphic curves.

Recall that the group of conformal automorphisms of the Riemann sphere S2 is the group of Moebius transformations, PSL(2, C), that from now on will be denoted just by G. We recall that its elements are of the form 32 CHAPTER 2. MODULI SPACES OF PSEUDOHOLOMORPHIC CURVES

az + b σ(z) = cz + d with ad − bc 6= 0. Recall also that this group acts transitively in the set of all three distinct points of S2. That is, 2 2 given distinct points z1, z2, z3 ∈ S and distinct points w1, w2, w3 ∈ S , there is a unique map 2 σ ∈ Aut(S ) satisfying σ(z1) = w1, σ(z2) = w2, and σ(z3) = w3.

If u : S2 −→ X is a pseudoholomorphic curve, we can consider the curves u ◦ σ for all conformal automorphisms ψ ∈ Aut(S2). This gives us an action of the group Aut(S2) on the moduli spaces M(A, J). We will be interested in the moduli spaces M(A, J)/G, quotient of M(A, J) by the action of this group. In the case that M(A, J) is a ﬁnite dimensional manifold, this quotient can be seen to be also a (ﬁnite dimensional) manifold.

Finally, we introduce the evaluation map, which is crucial for applications and further develop- ment of the theory, like the deﬁnition of Gromov-Witten invariants.

We have a natural evaluation mapev ˜ : M(A, J )×S2 −→ V deﬁned byev ˜ (f, z) = f(z). Consider now the action of G on M(A, J ) × S2 given by σ · (f, z) = (f ◦ σ−1, σ(z)). It is clear that the mapev ˜ factors through this action. If we consider the restriction to the moduli space M(A, J) for some almost complex structure J, we arrive to the next deﬁnition.

2 Definition 2.2.15. We call the evaluation map to the map evJ :(M(A, J) × S )/G −→ V defined by evJ ([f, z]) = f(z). Now, fix a regular J. Then, M(A, J) is a finite-dimensional manifold. It can be seen that in this situation, (M(A, J) × S2)/G is also a manifold.

2.3 Failure of compactness

In this section we will see a geometrically interesting example of moduli space of pseudoholomorphic curves where compactness fails. Considering S2 = CP 1 and CP 2 as K¨ahlermanifolds, with their standard complex structures (denoted by j and J respectively), we have that f : CP 1 −→ CP 2 is a pseudoholomorphic curve if it is a holomorphic curve, in the sense of complex geometry.

1 2 2 2 Consider the following sequence of maps un : CP −→ CP given by un([x : y]) = [x : y : nxy].

First of all, it is clear that un are holomorphic curves, hence they are J-holomorphic curves. We see that they all represent the same homology class. Indeed, they are all homotopic to the map 2 2 1 2 u0 deﬁned by u0([x : y]) = [x : y : 0] via the homotopy Hn : CP × I −→ CP given by

2 2 Hn([x : y], t) = [x : y : (1 − t)nxy]

We only have to check that Hn is well-deﬁned and continuous. Indeed, it is well-deﬁned since 2 2 2 1 [x : y : (1 − t)nxy] ∈ CP for all [x : y] ∈ CP and Hn([λx : λy], t) = Hn([x : y], t) for all λ ∈ C − {0}. The continuity is clear. Since the energy is a topological invariant, it follows 2 that all the un represent the same homology class, say A ∈ H2(CP ). Therefore, all of them are curves in the moduli space M(A, J), and represent elements in the moduli space M(A, J)/G. 2.3. FAILURE OF COMPACTNESS 33

0 0 We claim that there is no reparametrization un := un ◦ φn with φn ∈ G such that un has a convergent subsequence in the C∞ topology. Therefore the moduli space M(A, J)/G is non compact.

In order to see this, observe that the image of the un is the family of smooth algebraic curves of Z2 degree 2 (conics) deﬁned by XY = n . Observe that this family degenerates as n → ∞ to the pair of lines XY = 0. Since the reparametrization of the sequence un does not change its image, we have that there is no convergent subsequence of any reparametrized sequence. For otherwise, its limit would have image XY = 0, while there is no smooth parametrization f : CP 1 −→ CP 2 of this curve.

However, XY = 0 is the union of two lines, hence of two smooth holomorphic curves CP 1 −→ CP 2. Let us see that each of these lines arise as the limit of a suitable reparametrization of our 1 original sequence (except for the point [0 : 0 : 1]). Take φn([x : y]) = [x : ny] and observe that 1 1 1 1 1 φn ∈ G. Indeed, we have that φn(z) = nz. Then, un := un ◦ φn is deﬁned by un([x : y]) = [x2 : n2y2 : n2xy] = [x2/n2 : y2 : xy] which converges to u1([x : y]) = [0 : y2 : xy] = [0 : y : x] 2 which has as image the line X = 0. Similarly, if we take φn([x : y]) = [nx : y] then we have 2 2 2 2 2 2 2 2 2 2 un := un ◦ φn is deﬁned by un([x : y]) = [n x : y : n xy] = [x : y /n : xy] which has as limit the map u2([x : y]) = [x2 : 0 : xy] = [x : 0 : y] which has as image the line Y = 0.

We see here a phenomenon characteristic of pseudoholomorphic curves. We have seen that the original sequence has no convergent subsequence (in the C∞ sense), but suitable reparametriza- tions of the sequence have subsequences converging to pseudoholomorphic spheres, in such a way that the limit curve of the sequence is the union of this pseudoholomorphic spheres. A pseudoholomorphic sphere that develops in the limit by reparametrizing the sequence is called a bubble. We will formalize all this in the next section, and we will see that the only way in which compactness can fail is bubbling (formation of bubbles at certain points). This is the content of the Gromov compactness theorem. 34 CHAPTER 2. MODULI SPACES OF PSEUDOHOLOMORPHIC CURVES Chapter 3

The Gromov compactness theorem

In this chapter we state the Gromov compactness theorem in terms of Kontsevich’s stable maps. We will consider only pseudoholomorphic maps of genus 0, which are easier to describe.

In this chapter we follow closely [M-S1].

3.1 Moduli space of stable curves

As we have seen, the moduli spaces M(A, J)/G of pseudoholomorphic curves u : S2 −→ X are not compact in general. What we would like now is to compactify it in a geometrically meaningful way. The wanted compactification of M(A, J)/G will be denoted by M(A, J). In this section we will describe this compactification by first presenting the objects that we will consider, and then describing the topology we will consider in this set.

As a set, the points of M(A, J) will be equivalence classes of stable maps. Our ﬁrst task is to deﬁne them.

As we have noted in the last chapter, we will see that the only way in which compactness can fail is that a sequence of pseudoholomorphic maps develop bubbles at some points. Since a bubble is a pseudoholomorphic sphere, we will model our limiting objects as maps deﬁned over some set of spheres connected between themselves. Considering each sphere as a vertex of a graph, and an edge between vertices as indicating that the two spheres represented by the vertices are connected, it turns out that the adequate model will be that of a tree.

Deﬁnition 3.1.1. A graph G = (V,E) consists on a set V of vertices and a relation E (edges) in V which is symmetric and such that xEx/ for all x ∈ V .

Let x, y ∈ V . A path in G starting at x and ending at y is a ﬁnite sequence of vertices (v1, .., vn) such that v1 = x, vn = y and viEvi+1 for i = 1, ..., n − 1.

A graph G is said to be connected if for every pair of vertices x, y ∈ G there exists a path in G starting at x and ending at y.

35 36 CHAPTER 3. THE GROMOV COMPACTNESS THEOREM

A cycle in G is a ﬁnite sequence of vertices (v1, ..., vn) such that n ≥ 3, vi 6= vj for i 6= j, viEvi+1 for i = 0, ..., n − 1, and v1 = vn.

A ﬁnite graph G is said to be a tree if it is connected and has no cycles.

Given a tree T and two vertices α, β ∈ T with αEβ, we will denote by Tαβ the set of all vertices γ in T such that there exists a path from α to gamma passing trough β.

We will also need the notion of tree homomorphism.

Deﬁnition 3.1.2. Let T,T 0 be trees. A map f : T −→ T 0 is said to be a tree homomorphism if f −1(γ) is a tree for each vertex γ ∈ T 0, and for all α, β ∈ T with αEβ, either f(α)E0f(β) or f(α) = f(β).

If moreover f is bijective and f −1 is also a tree homomorphism, we say that f is a tree isomorphism.

Our stable maps will be modelled on trees. We deﬁne them now.

Deﬁnition 3.1.3. Let (X, ω) be a compact symplectic manifold and let J be an ω-compatible almost complex structure on X. A stable J-holomorphic map of genus zero into X, modelled over the tree T is a tuple

α (u, z) = ({u }α∈T , {zαβ}αEβ) α 2 2 where each u : S −→ X is a J-holomorphic curve and zαβ are points in S (called nodal points) labelled with the oriented edges αEβ of the tree T , such that the following conditions are satisﬁed:

α β 1. For all α, β ∈ T such that αEβ, we have that u (zαβ) = u (zβα).

0 2. For every α ∈ T , we have zαβ 6= zαβ0 if β 6= β .

We put Zα = {zαβ : αEβ}.

α 3. If u is a constant map, then #Zα ≥ 3.

Observe that this deﬁnition correspond with the intuitive image of the limit of a pseudoholomorphic curves, since as we have say, convergence can only fail via bubbling. Therefore, it is reasonable to expect that the limit is a map deﬁned on a bunch of spheres, one connected to others only through a point (the point where bubble develops) in such a way that there are no cycles, and such that the map restricted to each sphere is a pseudoholomorphic map, and the only point that two contiguous spheres have in common have the same image.

We have also an evident notion of energy for an stable map: just sum the energy of all the bubbles.

Deﬁnition 3.1.4. Let (u, z) be a stable map. Then we deﬁne its energy as X E(u) := E(uα) α∈T 3.1. MODULI SPACE OF STABLE CURVES 37

We also deﬁne

X γ mαβ(u) := E(u )

γ∈Tαβ In the similar way, stable maps represent second homology classes of X.

Deﬁnition 3.1.5. Let (u, z) be a stable map and A ∈ H2(X). We say that (u, z) represents the class A if

X α 2 A = u∗ ([S ]) α∈T We define the moduli space of stable curves representing the homology class A. Definition 3.1.6. The moduli space of J-stable curves representing the homology class A is defined by

SC(A, J) = {(u, z):(u, z) is a J-stable curve representing A} We have a natural notion of equivalence of stable curves by reparametrization, that we now describe. Deﬁnition 3.1.7. Two stable maps (u, z), (u0, z0), modelled over trees T,T 0 respectively, are called equivalent if there exists a tree isomorphism f : T −→ T 0 and a function φ : T −→ G (where we denote φ(α) by φα) such that for all α

0 1. uf(α) ◦ φα = uα

0 2. zf(α)f(β) = φα(zαβ) We deﬁne now the desired moduli spaces of pseudoholomorphic curves.

Definition 3.1.8. Let A ∈ H2(X), and J an almost complex structure on X. We define the moduli space of J-stable curves representing A, and denote it by M(A, J), as the quotient of SC(A, J) by the equivalence relation defined above (that is, equivalence of stable trees). We have to define a topology on the moduli space M(A). We will define the topology by defining first a notion of convergence of a sequence of stable maps. Definition 3.1.9. A sequence of stable maps

α n (un, zn) = ({un}α∈Tn , {zαβ}αEnβ) is said to Gromov converge to a stable map

α (u, z) = {u }α∈T , {zαβ}αEβ) if there exists a n0 such that for all n ≥ n0 there exists a surjective tree homomorphism

fn : T −→ Tn α and a collection of Moebius transformations {φn}α∈T , such that the following holds:

fn(α) α 2 1. For every α ∈ T the sequence of maps un ◦ φn : S −→ X converges uniformly on 2 compact subsets of S − Zα. 38 CHAPTER 3. THE GROMOV COMPACTNESS THEOREM

2. If αEβ, then:

α mαβ(u) = lim lim Efn(α)(un; φn(B(zαβ))) →0 n→∞

3. If α, β ∈ T are such that αEβ and nj is a subsequence such that fnj (α) = fnj (β) then φαβ := (φα )−1 ◦ φβ converges to z uniformly on compact subsets of S2 − {z . nj nj nj αβ αβ

4. If α, β ∈ T are such that αEβ and nj is a subsequence such that fnj (α) = fnj (β) then z = lim (φα )−1(znj ). αβ n→∞ nj fnj (α),fnj (β) Observe that this deﬁnition induces a notion of convergence in M(A, J).

Before going on, let us explain what this notion of convergence means. In order to do so, it will be helpful to write the definition of Gromov convergence in the more usual situation where the sequence of stable maps consist in pseudoholomorphic spheres. That is, the definition of convergence of a sequence of pseudoholomorphic spheres to a stable map. Definition 3.1.10. A sequence of pseudoholomorphic maps

2 un : S −→ X is said to Gromov converge to a stable map

(u, z) = {uα}α∈T , {zαβ}αEβ) α if there exists a collection of Moebius transformations {φn}α∈T , such that the following holds: α α 2 α (Map) For every α ∈ T the sequence of maps un := un ◦ φn : S −→ X converges to u 2 uniformly on compact subsets of S − Zα. (Energy) If αEβ, then:

α mαβ(u) = lim lim E(un; φn(B(zαβ))) →0 ν→∞

αβ α −1 β (Rescaling) If α, β ∈ T are such that αEβ then the sequence φn := (φn) ◦ φn converges to ∞ 2 zαβ in the C sense on S − {zβα}.

Observe that in the ﬁrst condition the sequence un does not enter directly, but only composed with Moebius transformations. This is because we are only interested in the limit of the sequence modulo the action of the reparametrization group G. Therefore, we do not ask the sequence to converge, but only a suitable reparametrization of the sequence. Taking this into account, the ﬁrst condition says that a suitable reparametrized sequence converge in the C∞ sense everywhere 2 on S except at the points of Zα, which are interpreted as the points where bubbling occurs in the limit.

The second condition asserts that there is no energy loss in the limit. Indeed, it says that the energy of the bubble tree bubbling at the point zαβ is all the energy concentrating at small neighbourhoods of the point zαβ. The fact that there is no energy loss in the limit implies that this is the correct notion of convergence, since we do not loss geometric information about the sequence when passing to the limit.

Now we can ﬁnally deﬁne the topology on the set of stable maps as follows. 3.2. STATEMENT OF THE COMPACTNESS THEOREM 39

Deﬁnition 3.1.11. A set C ⊂ M(A, J) is said to be Gromov closed if every Gromov convergent sequence of stable curves in C has a limit in C. A set U ⊂ M(A, J) is said to be Gromov open if its complement is Gromov closed. It can be checked that this deﬁnes a topology on M(A, J), called the Gromov topology.

3.2 Statement of the compactness theorem

In this section we will state and discuss the theorem we want to prove. As we have seen in the previous chapter, the moduli spaces M(A, J)/G are not compact in general. Since the compactness of a moduli space is a very desirable property (it is crucial for some applications, as we will see in the last chapter), we seek a compactiﬁcation of the moduli space. Moreover, we want a compactiﬁcation that is geometrically meaningful, so that the points we add to the moduli space gives us some insight about pseudoholomorphic curves. As said above, the new points we need to add to the moduli space are the so-called stable maps. They are geometrically meaningful, since they arise in a natural way as the limit of reparametrized sequences of pseudoholomorphic curves.

We state now a ﬁrst version of the Gromov compactness theorem that tells us that any sequence of pseudoholomorphic maps converge to a stable map.

Theorem 3.2.1 (Gromov compactness theorem). Let (X, ω) be a symplectic manifold and let ∞ Jn be ω-compatible almost complex structures converging in the C sense to an almost complex 2 structure J. Let un : S −→ X be a sequence of Jn-holomorphic curves with bounded energy. Then, there is a subsequence of the un that Gromov converges to a J-stable map (u, z). We can improve this theorem in order to prove that every sequence of stable maps of bounded energy has a subsequence that Gromov converges to a stable map.

Theorem 3.2.2 (Gromov compactness theorem for stable maps). Let (X, ω) be a symplectic ∞ manifold and let Jn be ω-compatible almost complex structures converging in the C sense to an almost complex structure J. Let (un,

Theorem 3.2.3. The moduli spaces M(A, J) are compact metrizable spaces. However, proving the metrizability of the moduli spaces would take us too far, and we will not do it in this work. For a proof, see chapter 3 of [M-S1]. 40 CHAPTER 3. THE GROMOV COMPACTNESS THEOREM Chapter 4

Local estimates

In this chapter we will study in detail the local behaviour of pseudoholomorphic curves. In particular, we will study the behaviour of pseudoholomorphic curves defined on disks and pseudoholomorphic curves defined on cylinders (which are equivalent to pseudoholomorphic curves defined on annuli), focusing on the distribution of the energy density along a sufficiently long cylinder.

After that, we will put to use the results we obtain for pseudoholomorphic cylinders in order to prove the removal of singularities theorem, which states that a pseudoholomorphic map defined on a punctured disk with bounded energy extends to a pseudoholomorphic map defined on the whole disk, and so it can be thought of as a pseudoholomorphic analogue of the Riemann extension theorem of complex analysis, which says that any bounded holomorphic function defined on a punctured disk extends holomorphically to the whole disk. We give here an elementary proof of this result, assuming only knwoledge of the usual L2 theory of linear elliptic operators.

The local estimates obtained in this chapter (and especially the removal of singularities theorem) constitute the core of our proof of the Gromov compactness theorem, that we will develop in the following chapter.

In all this chapter we denote by X a compact manifold endowed with an almost complex structure J. We consider X as embedded in some euclidean space RN , and take as metric in X the pullback of the euclidean metric in Rn. Observe that since X is compact, all the metrics in X induce equivalent norms.

We also assume in the following that all pseudoholomorphic curves are smooth.

4.1 Review of Sobolev spaces and elliptic regularity

In this section we collect the relevant deﬁnitions and theorems from L2 Sobolev theory that we need in the rest of the chapter. We will not present proofs, and refer the reader to Chapter 6 of [War].

We will consider functions ϕ : Rn −→ Cm. Moreover, in order to simplify things, and since we are only interested in the local behaviour of functions, we will assume that they are periodic of period 2π in each variable. This is not a restriction since we can always extend a function

41 42 CHAPTER 4. LOCAL ESTIMATES deﬁned in a neighbourhood of a point to a periodic function.

If we consider smooth periodic functions, we can develop them in Fourier series:

X ix·ξ ϕ(x) = ϕξe ξ n where ξ = (ξ1, ..., ξn) is an n-tuple of integers, and the sum is extended over Z . Recall also that the Fourier coeﬃcients are deﬁned by Z 1 −ix·ξ ϕξ = n ϕ(x)e (2π) Q where Q is a cube of side 2π.

We will identify a periodic function with the set of corresponding Fourier coefficients. This allows us to define Sobolev spaces of periodic functions. We denote by S the complex vector space of sequences of complex vectors in Cm indexed by n-tuples of integers ξ. Definition 4.1.1. For each integer k we define the k-Sobolev space as

2 X 2 k 2 Lk = {u ∈ S : (1 + |ξ| ) |uξ| < ∞} ξ and we endow it with the inner product deﬁned by

X 2 k < u, v >k= (1 + |ξ| ) uξ · vξ ξ

2 It can be seen that Lk are Hilbert spaces.

2 2 Observe that, for s ≥ 0, each u ∈ Ls represents an L function deﬁned by

X ix·ξ φ(x) = uξe ξ

Proposition 4.1.2. The space of all smooth periodic functions is dense in every Sobolev space 2 Lk. For a proof, see Theorem 6.18(c) of [War].

An interesting result, that gives some insight into the meaning of Sobolev spaces, is the following:

Proposition 4.1.3. Let k be a non-negative integer, and let ϕ be a smooth periodic function. 2 Then, the Sobolev norm in Lk is equivalent to the norm given by

k X ||ϕ|| = ||Dαϕ|| [α]=0 α where we use multi-index notation (α = (α1, ..., αn) and [α] = α1 + ... + αn), and ||D ϕ|| = α supx∈Q |D ϕ(x)|. 4.1. REVIEW OF SOBOLEV SPACES AND ELLIPTIC REGULARITY 43

For a proof, see Theorem 6.18(a) of [War].

Therefore, for smooth functions, the k-Sobolev norm takes into account the values of the function and the values of all the derivatives up to order k.

A key property of the Sobolev spaces is the fact that, if some function is in a suﬃciently high Sobolev space, then it is suﬃciently derivable. More precisely,

Theorem 4.1.4 (Sobolev lemma). If t > [n/2] + m + 1 (where n is the dimension of the space) 2 α P α ix·ξ and u ∈ Lt , then the series D u = ξ ξ uξe converges uniformly for [α] ≤ m. Therefore, 2 m each u ∈ Lt for this range of t corresponds to a function of class C . For a proof, see Lemma 6.22 of [War].

The other important theorem about Sobolev spaces is the following.

n 2 n Theorem 4.1.5 (Rellich lemma). Let u be a sequence of elements of Lt with ||u ||t ≤ 1. If n 2 s < t, then there is a subsequence of u which converges in Ls. In other words, the inclusion 2 2 i : Lt −→ Ls is compact. For a proof, see Lemma 6.33 of [War]. For the proof of the removal of singularities theorem, we will need the following lemma.

2 n Definition 4.1.6. Let u ∈ Lk and 0 6= h ∈ R . We define the difference quotient of u determined by h as

T (u) − u uh = h |h| 2 2 where Th : Lk −→ Lk is the operator deﬁned (using the Fourier coeﬃcients) by (Th(u))ξ := ih·ξ e uξ.

2 n h Lemma 4.1.7. Let u ∈ L . Then, for any 0 6= h ∈ , we have that ||u || 2 ≤ C||u|| 2 , k R Lk Lk+1 h where C does not depend on h. Conversely, if ||u || 2 ≤ K for some constant K independent of Lk 2 h, then u ∈ Lk+1. For a proof see 6.19 and 6.20 of [War].

Deﬁnition 4.1.8. A linear diﬀerential operator L of order l on the Cm-valued C∞ functions on n R consists of an m × m matrix (Lij) in which

l X α α Lij = aijD [α]=0 α n m ∞ α where aij : R −→ C are C and at least one aij 6= 0 for some i, j and some α with [α] = l. In principle, a differential operator is defined only for sufficiently differentiable functions. How- ever, since smooth functions are dense in all Sobolev spaces, it extends by density to an operator 2 2 L : Lk+l −→ Lk, where l is the order of L. 44 CHAPTER 4. LOCAL ESTIMATES

Deﬁnition 4.1.9. Let L be a diﬀerential operator of order l. We write L as

L = Pl(D) + ... + P0(D) where Pj(D) is an m × m matrix where each entry is a homogeneous diﬀerential operator or P α order j, [α]=j aαD .

n n We say that L is elliptic at x ∈ R if Pl(ξ) is non-singular at x for every 0 6= ξ ∈ R , where α α Pl(ξ) represents the matrix obtained from Pl(D) by replacing each occurrence of D with ξ .

We say that L is elliptic if L is elliptic at every x ∈ Rn. We have the following characterization of elliptic operators.

Proposition 4.1.10. L is elliptic if and only if L(φlu)(x) 6= 0 for each Cm-valued function u such that u(x) 6= 0 and each smooth real-valued φ such that φ(x) = 0 but dφ(x) 6= 0. See 6.28 of [War] for a proof.

We will be mainly concerned with the Cauchy-Riemann operator, which for a smooth function f : R2 −→ Cm is deﬁned as ¯ ∂f = ∂xf + i∂yf We have that

Proposition 4.1.11. The Cauchy-Riemann operator ∂¯ is elliptic.

Proof. Simply observe that ξ1 + iξ2 is never 0 for ξ = (ξ1, ξ2) 6= (0, 0). The following is the main inequality for elliptic operators. We will use it extensively in what follows

Proposition 4.1.12 (G˚arding’sinequality). Let L be an elliptic operator of order l, and let s be an integer. There is a constant C > 0 such that

||u|| 2 ≤ C(||Lu|| 2 + ||u|| 2 Ls+l Ls Ls 2 for all u ∈ Ls+l. See 6.29 of [War] for a proof. 2 Definition 4.1.13. Let L be a differential operator of order l, and v ∈ Ls for some s. Any 2 solution u ∈ Lk for some k of an equation Lu = v is called a weak solution of the differential equation. Such a weak solution is called a strong solution if u ∈ Cl. The main theorem about elliptic operators is the following.

Theorem 4.1.14 (Elliptic regularity). Let L be a periodic elliptic operator of order l. Assume 2 2 that u ∈ Lk for some k ∈ Z, v ∈ Lt and

Lu = v 2 Then, u ∈ Lt+l. For a proof, see 6.30 of [War]. 4.2. ESTIMATES ON DISKS 45

4.2 Estimates on disks

We start by proving some estimates for pseudoholomorphic maps deﬁned on disks.

2 The following lemma tells us that the Lk norms of the derivative of a pseudoholomorphic map on a small disk is controlled by the C0 norm of the derivative on a larger disk, provided the C0 norm of the derivative is small enough.

n Lemma 4.2.1. Let J be an almost complex structure in C satisfying J(0) = J0 (where J0 represents the canonical complex structure on the vector space Cn). For all k ∈ N, there exists n an > 0 such that if u : D1 −→ C is a J-holomorphic curve (i.e., it satisfy ∂J (u) = 0), satisfying |du| 0 < and |u(0)| < , then ||u|| 2 < Ck|du| 0 , for some constant C (D1) Lk(D1/2) C (D3/4) Ck > 0 independent of u.

2 Proof. First, observe that since u is pseudoholomorphic, it is smooth. Therefore, u ∈ Lk(D1) for all k. We prove the theorem by induction on k.

2 0 For k = 0, we have that ||u||L (D1/2) ≤ const|u|C (D1/2). R 1 But |u| 0 < 2 if |du| 0 < and |u(0)| < , since |u(z) − u(0)| ≤ |du| ≤ < C (D1/2) C (D1) [0,z] 2 1 and |u(z)| ≤ |u(0)| + |u(z) − u(0)| < 2, if |z| ≤ 2 . 2 0 Therefore, ||u||L (D1/2) ≤ const|du|C (D3/4), where the constant is independent of u.

Fix a k > 0 and suppose the lemma is true for this k.

By G˚arding’sinequality,

||u|| 2 ≤ const(||∂J u|| 2 + ||u|| 2 ) Lk+1(D1/2) 0 Lk(D1/2) Lk(D1/2)

Since 0 = ∂J u = ∂J0 + (J(u) − J0)∂yu we have ∂J0 u = −(J(u) − J0)∂yu.

Therefore:

||u|| 2 ≤ const(||(J(u) − J0)∂yu|| 2 + ||u|| 2 ) Lk+1(D1/2) Lk(D1/2) Lk(D1/2)

r We now want to estimate ||(J(u)−J0)∂yu|| 2 . In order to do that, observe that ∇ ((J(u)− Lk(D1/2) P p q J0)∂yu) = p+q=r ∇ (J(u) − J0)∇ ∂yu. Since both J and u are smooth, so is J(u) − J0.

R Observe that for z ∈ D we have |u(z) − u(0)| ≤ |du| < if |du| 0 < . Then, 1/2 [0,z] 2 C (D3/4) |u(z)| ≤ |u(0)| + /2 < + < 2. Therefore, we can ensure that u(D1/2) ⊂ B2 if we have |du| < and |u(0)| < , choosing > 0 small enough.

k Let δ > 0, an choose > 0 so that it also satisﬁes ||J||C (B2) < δ (such a ball B2 exists since J is smooth).

For p, q < k, we have, by induction hypothesis, 46 CHAPTER 4. LOCAL ESTIMATES

p q p p q 2 2 ||∇ (J(u) − J0)∇ ∂xu||L (D1/2) = ||(∇ (J − J0)) ◦ u∇ (u)∇ ∂xu||L (D1/2) p p q+1 0 2 ≤ ||(∇ (J − J0)) ◦ u||C (D1/2)||∇ (u)∇ (u)||L (D1/2) p q+1 2 ≤ δ||∇ (u)∇ (u)||L (D1/2)

2 Observe now that, by the Sobolev embedding theorem, we have that if ||u||L (D1/2) < then k−2 2 |u|C ≤ ||u||L (D1/2), if k ≥ 2.

Since p + q = r ≤ k, either p < k − 1 or q + 1 < k − 1, if k > 3. In this case, if p < k − 1, we obtain

p q q 2 p 2 ||∇ (u)∇ (u)||L (D1/2) ≤ |u|C (D1/2)||∇ (u)||L (D1/2)

≤ |u| p ||u|| 2 ≤ C||u|| 2 C (D1/2) Lk+1(D1/2) Lk+1(D1/2)

k−2 2 where in the last step we have used |u|C ≤ ||u||L (D1/2), the induction hypothesis, and we 0 assume that |du|C (D1/2) < 1. We get the same result if q + 1 < k − 1. In the cases k = 1, 2, 3, observe that we always have p ≤ 1 or q ≤ 1. Then, we can use the fact that we may assume p q |du| 0 < 1 and |u| 0 ≤ 2, to obtain in all the cases ||∇ (u)∇ (u)|| 2 ≤ C||u|| 2 , C C L (D1/2) Lk+1(D1/2) where C is a constant independent of u.

r Therefore, we conclude that ||∇ ((J(u) − J0)∂xu)|| 2 ≤ constδ||u|| 2 , if r < k, L (D1/2) Lk+1(D1/2) where the constant is independent of u.

Putting all together, we obtain ||R(u)|| 2 = ||(J(u)−J0)∂xu|| 2 ≤ C(δ)||u|| 2 , Lk(D1/2) Lk(D1/2) Lk+1(D1/2) where C(δ) is a constant depending only on δ. Looking at the deﬁnition of δ, we can take small

0 enough such that if |du|C (D1) < , then const.C(δ) < 1/2.

Then, returning at the G˚arding’sinequality, we get

||u|| 2 ≤ 1/2||u|| 2 + const||u|| 2 Lk+1(D1/2) Lk+1(D1/2) Lk(D1/2)

So, ||u|| 2 ≤ 2const||u|| 2 . And, by induction hypothesis, Lk+1(D1/2) Lk(D1/2)

||u|| 2 ≤ Ck|du| 0 Lk(D1/2) C (D1)

0 if |du|C (D1) is small enough. Finally, we conclude that there exists an > 0 and a constant Ck > 0 such that if |du| 0 < , ||u|| 2 ≤ Ck|du| 0 , as wanted. C (D1) Lk+1(D1/2) C (D1)

2 0 We can now prove that the Lk norms of du are controlled by the C norm of du.

Corollary 4.2.2. Let u : D1 −→ X be a pseudoholomorphic map. Then, for each k ∈ N there is some > 0 and a constant Ck > 0, such that if |du| 0 < , then ||du|| 2 ≤ C (D1) Lk(D1/2) 0 Ck|du|C (D3/4), where and Ck are independent of the map u. Proof. Fix a k. Let δ > 0 be the number corresponding to k in the preceding proposition. By the compactness of X, we can ﬁnd a number > 0, a ﬁnite open cover of X, U1, ..., Un, and 0 charts ξi : Ui −→ D1 such that for any pseudoholomorphic map u : D1 −→ X with |du|C (D1) < 4.2. ESTIMATES ON DISKS 47

−1 0 0 0 there exists an i with u(D1) ⊂ ξi (D1/2) and, if we deﬁne u = ξi ◦ u, then |du |C (D1) < δ and |u0(0)| < δ.

0 0 Then, by the previous lemma, ||u || 2 ≤ C|du| 0 . In particular, ||du || 2 ≤ Lk+1(D1/2) C (D3/4) Lk(D1/2) 0 0 ||u || 2 ≤ C|du | 0 . Lk+1(D1/2) C (D3/4)

−1 Now, observe that the norm induced by the norm of X in ξi (D1/2), and the norm induced by −1 the pullback of the standard metric of D1 by ξi in ξi (D1/2) are equivalent. Since this is true for any i and there are a ﬁnite number of such i’s, there exist a constant C > 0 (independent of 1 0 0 u) such that if u satisﬁes the above conditions, then C |du(z)| ≤ |du (z)| ≤ C|du (z)|. Therefore, the result for u follows from that of u0. With this result at our disposal, we can prove a compactness result for sequences of pseudoholomorphic curves with bounded energy density.

Proposition 4.2.3. Let un :Σ −→ X be a sequence of Jn-holomorphic curves, and suppose that ∞ the almost complex structures Jn converge to an almost complex structure J in the C sense. Let ˚0 0 0 K ⊂ K ⊂ K ⊂ Σ, where K,K are two compact sets. Assume also that supn supx∈K0 |dun(x)| <

C < ∞, for some C > 0. Then, there exists a subsequence (unk )k that converges to a J- holomorphic curve u in C∞(K). Proof. We start by showing that there is a subsequence that converges in Ck(K), for every k. By the Sobolev embedding theorem, it is enough to check that there is a convergent subsequence 2 in Lk(K) for any natural k.

2 Fix a natural k. We want to ﬁnd a convergent subsequence of un in Lk(K). Let k+1 be the 0 coming from lemma 4.2.1 for k + 1. Choose, for each x ∈ K, an open set Ux ⊂ K that is the domain of a chart χx : Ux −→ C centered at x.

x For each open set Ux, put un = un|Ux ◦ χx. Without loss of generality, we may assume that x 2 2 k+1 |dun|C0 < k+1 (if not, replace χx by χx ◦ ψ, where ψ : R −→ R is deﬁned by ψ(z) = C z).

i Since X is compact, passing to a subsequence we may assume that the sequence (un(0))n converges to a point x ∈ X. Pick a chart (V, ξ) of X centred at x and such that J(0) = J0 in this i i 0 chart. For n big enough we have that ξ(un(0)) ∈ Dk+1 . Since also |dun|C (D1) < k+1, by the i lemma above, we get that ||u || 2 ≤ Ak+1, if we can obtain an k+1 that works for all n Lk+1(D1/2) almost complex structures Jn, for n big enough.

Indeed, ﬁxed a δ > 0, for n big enough, we have ||J(x) − Jn(x)|| 2 < δ (by the Sobolev Lk(Dr ) ∞ embedding theorem together with the fact that Jn converges to J in the C sense). Then, for n big enough and any > 0, if ||J − J0|| 2 < − δ, we have ||Jn − J0|| 2 < Lk(Dr ) Lk(Dr ) δ + ||J − J0|| 2 < for n big enough. By looking at the proof of lemma 4.2.1, we see that Lk(Dr ) this is enough to see that the lemma applies to the present situation and we can take k+1 to be the same for all the almost complex structures Jn. 2 Hence, by Rellich’s lemma, it has a convergent subsequence in Lk.

2 Therefore, for any Ux we have a subsequence of un that converges to a Lk function ux on Ux. Since K is compact and the {Ux}x∈X is an open covering of K, we can ﬁnd a ﬁnite subcover 2 U1, ..., Un of K. Then, we can obtain a subsequence of un that converges in Lk to a function u in 48 CHAPTER 4. LOCAL ESTIMATES

1 all of K. Indeed, we can extract a convergent subsequence un that converges in U1. Then from 2 this sequence we can extract a subsequence un that converges in U2, and continue until we get 2 a subsequence that converges in Lk(K). Finally, we repeat this process with respect to k, that 1 2 is, we choose a subsequence un of the original sequence that converges in L1(K), extract from 2 2 this another subsequence un that converges in L2(K) and proceed recursively for each k. Finally n consider the diagonal sequence (un)n. By construction, this is a subsequence of our original 2 sequence that converges in Lk(K) for every k. By the Sobolev embedding theorem, we have that this sequence in fact converges in Cn(K) for every n, or in other words, that it converges ∞ ∞ in C (K). Call u the limit of this sequence. Then, u ∈ C (K). Moreover, u satisﬁes ∂J u = 0. n n Indeed, we have that ∂Jn un = 0 for each n, and un converges to u uniformly, so taking the limit as n goes to ∞, we obtain that u is a J-holomorphic curve, as wanted.

To ﬁnish, we prove a converse of corollary 4.2.2, namely that we can control the C0 norm of du on some disk if we have control over the L2 norm on some larger disk, provided the energy of the curve is small enough. But ﬁrst, we need the following lemma.

Lemma 4.2.4. Let u : D1 −→ X be a J-holomorphic map. Then there exists an > 0 such that 2 2 if ||du||L (D1) < then |du(0)| ≤ K||du||L (D1) for some constant K > 0 independent of u. Proof. Suppose that the conclusion of the statement is not true. In this case, we can ﬁnd a se- |duj (0)| quence of J-holomorphic maps uj : D1 −→ X such that ||duj|| 2 → 0 and limj = L (D1) ||duj || 2 ∞. L (D1)

We will split the proof in two cases: that |duj(0)| → 0, and that there is some > 0 such that |duj(0)| ≥ for all j.

Case 1: |duj(0)| → 0.

We will see that we can reduce this case to the second one. Indeed, passing to a subsequence, we may assume that the images of all the uj are contained in one coordinate chart of X, by taking the of the statement small enough. Let ξ : U −→ X be this chart, and we denote by J again 2n 0 the almost complex structure ξ∗(J) in R induced by J via ξ. Consider uj = ξ ◦ uj. Then, also 0 0 |duj (0)| |du (0)| → 0 and lim 0 = ∞. j j ||du || 2 j L (D1) 0 −1 2 Let, for each j, ψj(z) = |duj(0)| uj(z). Then, |dψj(0)| = for all j, but ||dψj||L (D1) = 0 −1 0 2 |duj(0)| ||duj||L (D1) → 0. Moreover, ψj satisﬁes ∂Jj ψj = 0, where Jj is the almost complex 2n 0 −1 0 structure in R induced by J via the map x 7→ |duj(0)| ξ(x). Observe that, since |duj(0)| → 0, ∞ the almost complex structures Jj converge to J0 in the C sense.

Case 2: There is some > 0 such that |duj(0)| ≥ for all j.

First of all, we prove that if f : D1 −→ R≥0 is a smooth function satisfying f(0) > 0, then there k−1 exists some integer k ≥ 1 and a point x ∈ D1 such that f(x) ≥ 3 f(0), D3−k (x) ⊂ D1, and sup < 3f(x). D3−k (x) Indeed, we can deﬁne in a recursive way a sequence of points x0, x1, ... ∈ D1 satisfying |xj − −j xj−1| ≤ 3 for any j, as follows. Let x0 = 0 and suppose x0, x1, ..., xi−1 have been deﬁned. By hypothesis, 4.2. ESTIMATES ON DISKS 49

−(i−1) −(i−2) −1 |xi−1 − x0| ≤ |xi−1 − xi−2| + |xi−2 − xi−3| + ... + |x1 − x0| ≤ 3 + 3 + ... + 3 < P −j j≥1 3 = 2/3 so D3−i (xi−1) ⊂ D1. Choose xi ∈ D3−i (xi−1) in such a way that f(xi) = sup{f(y)|y ∈ D3−i (xi−1)}. For some j ≥ 1 we must have f(xj) < 3f(xj−1). Indeed, if this is not the case, we would have j f(xj) ≥ 3 f(0) for every j (using induction). Since f(0) > 0 and we have seen that |xj| ≤ 2/3 for every j, we have a contradiction with the fact that f| is bounded. Let k be the smallest D2/3 j ≥ 1 such that f(xj) < 3f(xj−1), and put x = xj. These x and k satisfy the required properties.

With this, we can show that the assumption leads to a contradiction. In order to handle the case 1, we allow the maps uj to be Jj-holomorphic maps where Jj are diﬀerent almost complex structures that converge to an almost complex structure J in the C∞ sense.

Suppose that the statement is false. Then, there is a sequence of pseudoholomorphic maps |duj (0)| uj : D1 −→ X such that ||duj|| 2 → 0 and limj = ∞. L (D1) ||duj || 2 L (D1)

We prove ﬁrst that for any > 0 there is a C > 0 such that if u : D1 −→ X is pseudoholomorphic 2 and ||du||L (D1) < C then |du(0)| < . By the result just proved, for each j there are xj ∈ D1 and kj ∈ N such that:

k k j j 2 1. |duj(xj)| ≥ 3 |duj(0)| ≥ 3 j||duj||L (D1)

2. there is an inclusion D3−j (xj) ⊂ D1

3. the number µj = supD −k (xj )|duj| satisﬁes µj ≤ 3|duj(xj)| 3 j

−kj −1 Deﬁne, for each j, ρj = min{3 , µj δ}, where δ is the number C coming from 4.2.3. 2 Deﬁne now maps ψj : D1 −→ X, by ψj(z) = j||duj||L (D1)u(xj +ρjz), which make sense because −kj ρj ≤ 3 and then xj + ρjz ∈ D3−kj ⊂ D1.

We have:

∂I ψj = 0

supD1 |dψj| ≤ ρj sup |duj| = ρjµj ≤ δ D −k (xj ) 3 j

2 2 2 ||dψj||L (D1) ≤ ||duj||L (D −k (xj )) ≤ ||duj||L (D1) → 0 3 j Then,

−kj −1 −kj −1 |duj(0)| = ρj|duj(xj)| = min{3 , µj δ}|duj(xj)| = min{3 |duj(xj)|, µj δ|duj(xj)|}

−kj −1 −1 We can estimate 3 |duj(xj)| ≥ and µj δ|duj(xj)| ≥ 3 δ which implies that |dψj(0)| ≥ min{, 3−1δ}.

1 By proposition 4.2.3 there exists a subsequence ψjk that converges in the C sense uniformly on 2 D1/2 to a map ψ : D1/2 −→ X. Then, by the above estimates, we have that ||dψ||L (D1/2) = 0, 50 CHAPTER 4. LOCAL ESTIMATES

−1 so dψ is identically 0. But we have seen also that |dψj(0)| ≥ min{, 3 δ} which implies that |dψ(0)|= 6 0, a contradiction.

Corollary 4.2.5. Let u : D1 −→ X be a J-holomorphic map. Then there exists an > 0 such 2 0 2 that if ||du||L (D1) < then |du|C (D1/2) ≤ K||du||L (D1) for some constant K > 0 independent of u.

Proof. Consider, for each w ∈ D1/2, the map uw : D1 −→ X deﬁned by uw(z) = u(w + z/4). This is well deﬁned because |w + z/4| ≤ 1/2 + 1/4 < 1. Moreover, |duw(0)| = |du(w)|/4, and 2 2 ||duw||L (D1) ≤ ||du||L (D1) < .

We can therefore apply the previous lemma to each of the maps uw in order to obtain |du(w)|/4 = 2 2 2 |duw(0)| < K||duw||L (D1) ≤ K/4||du||L (D1), and therefore, |du(w)| < K||du||L (D1) for all w ∈ D1/2, as wanted.

4.3 Estimates on cylinders

We now study the behaviour of pseudoholomorphic curves deﬁned on cylinders. The goal of this section is to prove that in a ﬁnite cylinder, the energy density decays expo- nentially when we move towards the center of the cylinder, where these estimates are given by constants which are independent of the length of the cylinder. This estimate will be essential in our proof of the removal of singularities theorem (see theorem 4.4.1), and also in the analysis of bubbling that we will undertake in the following chapter.

1 1 R 2 For the next lemma, let Ei(a) := E(u;[a(i − 1), ai] × S ) = 2 [a(i−1),ai]×S1 |du(t, θ)| .

Lemma 4.3.1. Let a > 0 be a real number, and let u : S1 × [0, 3a] −→ Cn be holomorphic (where we consider the standard complex structure on the cylinder). Then, we have E2(a) ≤ γ −1 2 (E1(a) + E3(a)) for some γ < 1. Moreover, any cosh (2a) < γ < 1 works.

Proof. First, suppose that n = 1. Then, u : S1 × [0, 3a] −→ C is holomorphic. We can develop P ikθ u as a Fourier series in the θ variable: u(θ, t) = uk(t)e . Since u is holomorphic, it k∈Z must satisfy the Cauchy-Riemann equations: ∂θu = i∂tu. If we compute it with the Fourier P ikθ P 0 0 series expression, we obtain: uk(t)ike = i u (t). We must then have u = kuk k∈Z k∈Z k k kt for all k ∈ Z. Solving this diﬀerential equation, we see that uk(t) = Cke , and we have P kt ikθ u(θ, t) = Cke e . k∈Z

P kt ikθ 2 Now, ∂tu(θ, t) = Ckke e . From the Cauchy-Riemann equations, we have |du| = k∈Z 2 2 2 1 R 2π 2 |∂tu| + |∂θu| = 2|∂tu| . If we deﬁne the energy at t as E(t) = 2 0 |du(θ, t)| dθ, we get

Z 2π X ∗ 0 (k+k0)t i(k−k0)θ X 2 2 2kt E(t) = CkCk0 kk e e = 2π |Ck| k e 0 0 k,k ∈Z k∈Z Finally, we have 4.3. ESTIMATES ON CYLINDERS 51

Z ia Z ia X 2 2 2kt Ei(a) = E(t)dt = 2π |Ck| k e dt (i−1)a (i−1)a k∈Z X 2 2kai 2ka(i−1) = π |Ck| k(e − e ) k∈Z,k6=0 γ Then, there exists a γ with 0 < γ < 1 that satisﬁes E2(a) ≤ 2 (E1(a) + E3(a)) if and only if 4ka 2ka γ 2ka 6ka 4ka γ 4ka 2ka 2ka −2ka e − e ≤ 2 (e − 1 + e − e ) = 2 (e − e )(e − e ) for all k, if and only if 1 ≤ γ cosh(2ka) for all k 6= 0, if and only if γ ≥ cosh−1(2ka) for all k 6= 0. Clearly, the worst case is k = 1, so it is enough to take γ with cosh−1(2a) < γ < 1.

The general case follows easily from the n=1 case. Indeed, let u : S1×[0, 3a] −→ Cn. We can write 1 then u = (u1, ..., un), where uj : S × [0, 3a] −→ C are holomorphic for j = 1, ..., n. Therefore, 1 R ai R 2π 2 1 Pn R ai R 2π 2 Pn j Ei(a) = 2 a(i−1) 0 |du(θ, t)| dtdθ = 2 j=0 a(i−1) 0 |duj(θ, t)| dtdθ = j=0 Ei (a). Hence, Pn j γ Pn j j γ E2(a) = j=0 E2(a) ≤ 2 j=0(E1(a) + E3(a)) = 2 (E1(a) + E3(a)), as wanted.

Proposition 4.3.2. Let a > 0, η > 0 and let u : S1 ×[−η, 3a+η] −→ X be a pseudoholomorphic −1 γ curve. Then, for any γ ∈ (cosh (2a), 1), there exists an > 0 such that E2(a) ≤ 2 (E1(a) + E3(a)) whenever |du|C0 < . Proof. Suppose that this is not the case. Then, there exists a sequence of pseudoholomorphic 1 j γ j j maps uj : S × [−η, 3a + η] −→ X with |duj|C0 → 0 and E2(a) > 2 (E1(a) + E3(a)), where j 1 Ei (a) := E(uj; [(i − 1)a, ia] × S ).

Since |duj|C0 → 0, we can assume without loss of generality (maybe passing to a subsequence) that the image of the uj’s is contained in a coordinate chart of X. In other words, we can assume 1 2n 2n that in fact uj : S × [−η, 3a + η] −→ R , where we have in R the almost complex structure J induced from the manifold X (that will not coincide, in general, with the standard complex 2n structure J0 of R ). Without loss of generality we can assume that J(0) = J0.

0 0 0 Consider now maps uj deﬁned by uj(z) = uj(z)/|duj|C0 . Then, |duj|C0 = 1, so the energy density of the sequence remains bounded. Moreover, this maps are pseudoholomorphic maps with respect to almost complex structures Jj satisfying Jj(z) = J(|duj|C0 z). Since |duj|C0 → 0, J(0) = J0, and the smoothness of the almost complex structure, we obtain that in fact Jj ∞ converges to J0 in the C sense. 0 j γ j j Finally, note that the maps uj satisfy E2(a) > 2 (E1(a) + E3(a)).

0 1 n Now, by 4.2.3, we can extract a subsequence of uj that converge to a map u : S × [0, 3] −→ C which is holomorphic (which is the same as J0-holomorphic). By the above remarks, it satisfy γ γ C E2(a) ≥ 2 (E1(a)+E3(a)) > 2 (E1(a)+E3(a)). But this is a contradiction with lemma 4.3.1. The following lemma will give us the link between the convexity relations we have found in the above proposition and the exponential decay behaviour of the energy density.

Lemma 4.3.3. Let e0, e1, ..., eN nonnegative real numbers. Assume that there is γ ∈ (0, 1) such γ that ej ≤ 2 (ej−1 + ej+1) for all k = 1, ..., N − 1. Then, there exists a σ > 0 and a constant C (independent of N) such that ej ≤ C(e0 + eN ) exp(−σ min{j, N − j}), for all j = 1, ..., N − 1. 52 CHAPTER 4. LOCAL ESTIMATES

γ Proof. First, suppose that we have ej = 2 (ej−1 + ej+1) for all k = 1, ..., N − 1. In this case, we can solve explicitly the recurrence equation, and we can obtain an expression for ej in terms of e0 and eN . The solution is

N N e0r+ − en j en − e0r− j ej = N N r− + N N r+ r+ − r− r+ − r−

1 p 2 1 p 2 where r+ = γ (1 + 1 − γ ) and r− = γ (1 − 1 − γ ). Observe that, since γ < 1, r+ > 1 and −1 r− < 1, and note also that r− = r+.

Now,

N −N e0r+ − en e0 − enr+ 1 N N = N ≤ (e0 + eN ) r+ − r− 1 − (r−/r+) 1 − (r−/r+) On the other hand,

N N en − e0r− en − e0r− −N 1 −N N N = N r+ ≤ (e0 + eN )r+ r+ − r− 1 − (r−/r+) 1 − (r−/r+) Therefore, putting C = 1 , we get 1−(r−/r+)

j −(N−j) −j −(N−j) ej ≤ C(e0 + eN )(r− + r+ ) = C(e0 + eN )(r+ + r+ )

≤ 2C(e0 + eN ) exp(−σ min{j, N − j}) where σ = log(r+).

γ For the general case, given e0, eN we pute ˜j the solution of ej = 2 (ej−1 + ej+1) for all k = 1, ..., N − 1. We deﬁne rj =e ˜j − ej. It is enough to see that rj ≥ 0 for all j. This is true for j = 0,N (since e0 =e ˜0 by hypothesis, and similarly with N, so r0 = rN = 0).

γ Observe that the rj’s satisfy rj ≥ 2 (rj−1 + rj+1) for j = 1, ..., N − 1.

Suppose that for some j > 0 we have rj < 0, and let j be the ﬁrst one for which this happens. We will prove that in this case, we have rk < 0 for all k ≥ j, which is a contradiction with the fact that rN = 0. γ γ γ Since 0 > rj, we have 0 > rj ≥ 2 (rj−1 + rj+1) ≥ 2 rj+1, so we have 0 > rj ≥ 2 rj+1, and in (n+1)γ particular, rj+1 < 0. We prove by induction on n that 0 > rj+n ≥ n+2 rj+n+1. We have just proved the case n = 0. Suppose now that this holds for n. Then,

γ γ (n + 1)γ2 γ r ≥ r + r ≥ r + r j+n+1 2 j+n 2 j+n+2 2(n + 2) j+n+1 2 j+n+2

(n+1)γ2 γ So we have 1 − 2(n+2) rj+n+1 ≥ 2 rj+n+2. But observe that since γ < 1, we have

(n + 1)γ2 n + 1 n + 3 1 − ≥ 1 − = 2(n + 2) 2(n + 2) 2(n + 2) and since rj+n+1 < 0, 4.3. ESTIMATES ON CYLINDERS 53

(n + 1)γ2 n + 3 1 − r ≤ r 2(n + 2) n+j+1 2(n + 2) n+j+1 Finally, we get

n + 3 γ r ≥ r 2(n + 2) n+j+1 2 j+n+2 and

(n + 2)γ r ≥ r n+j+1 n + 3 j+n+2

Since 0 > rn+j+1, it follows that 0 > rj+n+2 and the inductive hypothesis holds.

This ﬁnishes the proof of the lemma.

This last result, together with 4.3.2 implies that if u : C −→ X is a pseudoholomorphic curve, then for every k has an exponential decay with t for t large enough. This is the content of the next proposition.

Proposition 4.3.4. Let u : C −→ X be a pseudoholomorphic curve with domain a semiinﬁnite cylinder C = [0, +∞) × R, and suppose that u has ﬁnite energy. Then, there is some T > 0 such k that |∇ u(t, θ)| ≤ Ck exp(−σt) for any t > T , where we can take, for any > 0, σ < 1 − . Proof. Fix > 0. By proposition 4.3.2 and lemma 4.3.3, we obtain that, for any a > 0, 0 0 1 p 2 Ej(a) ≤ C exp(−σ j) for j big enough, where σ = γ (1 + 1 − γ ), and γ any real number 1 satisfying cosh(2a) < γ < 1.

1 −2a(1−) Observe that for a > 0 small enough, we have cosh(2a) < e . Taking such an a, we can 1 −2a(1−) 0 1 take a γ such that cosh(2a) < γ < e . Then we have σ > log γ > 2a(1 − ).

2 2 σ0 Note now that Ej(a) = ||du[(j−1)a,ja]||L2 , so in fact we obtain ||du[(j−1)a,ja]||L2 ≤ C exp(− a ja) ≤ 0 C exp(−2(1 − )ja) and ﬁnally ||du[(j−1)a,ja]||L2 ≤ C exp(−(1 − )t), for any t ∈ [(j − 1)a, ja] and j big enough. This means that we can take σ > 1 − .

Fix a δ > 0. Since the energy of u is ﬁnite, there is some T > 0 such that the energy of u|[T,+∞)×S1 is less than δ > 0.

Now we want to apply corollary 4.2.5 in order to obtain a bound

0 0 |du|C0([(j−1)a,ja]) < C exp(−σ j) 1 Indeed, we can cover [−η + (j − 1)a, ja + η] × S by ﬁnitely many open sets U1, ..., Un in the −1 cylinder and charts ξi : Ui −→ D1 such that ξi (D1/2) also cover the cylinder.

Deﬁne now uj : D1 −→ X by uj = u ◦ ξi. Choosing appropriately the charts ξi, we can assume 2 0 2 that ||duj||L (D1) < δ. By 4.2.5, we have then that |duj|C (D1/2) ≤ C||duj||L (D1). 54 CHAPTER 4. LOCAL ESTIMATES

So we obtain |du| −1 ≤ Cj||du|| 2 −1 , where in general the Cj is diﬀerent for C0(ξj (D1/2)) L (ξj (D1/2)) 0 each j, since it depends on the C norm of dξj. Since this is true for every j, choosing C = max{C1, ..., Cn} we ﬁnally obtain our desired bound

|du|C0([(j−1)a,ja]) ≤ C||du||L2([(j−1)a,ja]) Therefore for any t ∈ [(j − 1)a, ja] we have

σ0 |du(t, θ)| ≤ C0 exp(− t) ≤ C0 exp(−σt) a where σ > 1 − for t > T > 0.

We now show the analogous C0 bound for ∇ku. Repeating the same argument as above, we see by corollary 4.2.2, that, for any k,

00 ||du|| 2 < C exp(−σt) Lk([(j−1)a,ja]) Now, applying Sobolev lemma, we get our desired conclusion,

00 |du|Cr ([(j−1)a,ja]) < C exp(−σt) In particular, we get the result of the statement.

We ﬁnish this section by proving a result about annuli that we will need in the proof of the Gro- mov compactness theorem. Recall that an annulus is conformally equivalent to a ﬁnite cylinder, therefore we can apply the methods of this section.

We deﬁne A(r, R) := BR − Br, with r < R. Observe that A(r, R) is conformally equivalent to 1 the cylinder Cr,R := [0, log R − log r] × S via the conformal map ψ : Cr,R −→ A(r, R) deﬁned by ψ(t, θ) = et+log r+iθ.

Lemma 4.3.5. For every σ < 1 there exist constants > 0 and C > 0 such that for every pseudoholomorphic map u : A(r, R) −→ X with E(u) < we have, for all T such that eT r < e−T R:

E(u; A(eT r, e−T R)) ≤ Ce2σT E(u) Proof. Put u0 : C −→ X the map u0 := u ◦ ψ. Since the energy is conformally invariant, and we have ψ(log r + T, θ) = eT eiθ and ψ(log R − log r − T, θ) = Re−T eiθ, we obtain that

E(u; A(eT r, e−T R) = E(u0;[T, log R − log r − T ] × S1) Now, pick a > 0 so that T = ka,(k0 − 1)a ≤ log R − log r − T < k0a, with k0 > k, and (N − 1)a ≤ log R − log r < Na. Then, using lemma 4.3.3, we get:

k0−1 0 1 X E(u ;[T, log R − log r − T ] × S ) ≤ Ej(a) j=k k0−1 X ≤ C(E1(a) + EN (a)) exp(−2σa min{j, N − j}) j=k 4.4. REMOVAL OF SINGULARITIES 55 if E(u) < , by taking > 0 as in proposition 4.3.2, and a > 0 small enough as in the proof of proposition 4.3.4 to get the inequality with the desired σ.

0 T 0 log R−log r−T Observe that E1(a) + EN (a) ≤ E(u ) = E(u). Observe also that k = a and k = a = T T 0 N − a , so min{j, N − j} ≤ a for all k ≤ j ≤ k .

So we conclude

E(u0;[T, log R − log r − T ] × S1) ≤ C0E(u)e2σT as wanted.

4.4 Removal of singularities

In this section we will prove the theorem of removal of singularities, which is a key theorem in the proof of the Gromov compactness theorem.

∗ In this section we put D = D1 and D = D − {0}.

∗ Theorem 4.4.1 (Removal of singularities). Let u0 : D −→ X be a pseudoholomorphic map with ﬁnite energy. Then, u extends to a pseudoholomorphic map u : D −→ X.

Recall that C = [0, +∞) × S1 and D∗ are conformally equivalent. A conformal map between them is h : C −→ D∗ given by h(t, θ) = e−t+iθ.

∗ Proposition 4.4.2. Let u0 : D −→ X be a pseudoholomorphic curve with ﬁnite energy. Then, u0 extends to a continuous map u : D −→ X.

Proof. Since D∗ is conformally equivalent to C = [0, +∞) × S1 and the energy is a conformal 0 invariant, we can see u0 as a pseudoholomorphic curve u0 : C −→ X with ﬁnite energy (deﬁned 0 0 by u = u◦h). Applying proposition 4.3.4, we have that |du0(t, θ)| < C exp(−σt), for some σ > 0.

From this, we get that u0 has a continuous extension to all of D. Indeed, suppose that xk is a −1 sequence of points in D such that xk → 0. Write (tk, θk) = h (xk). Then, we have

Z 0 0 0 |u0(xp) − u0(xq)| = |u0(tp, θp) − u0(tq, θq)| ≤ |du (t, θ)| γ Z Z tq ≤ C exp(−σt) = C exp(−σt) < γ tp

R ∞ if p, q > N for some N, because 0 exp(−σt)dt is convergent. This shows that the sequence xk is Cauchy, and using the fact that X is complete (because it is compact), we obtain that u0(xk) is convergent to a point x ∈ X. Moreover, the argument also shows that the limit is independent of the sequence (if this were not the case, we could choose a sequence xk → 0 that is not Cauchy). Therefore, we can deﬁne u : D −→ X extending u0 continuously by putting u(0) = x. 56 CHAPTER 4. LOCAL ESTIMATES

Now that we have that u0 has a continuous extension to D, we need to show that this extension is in fact C∞. We will do it by proving a regularity theorem. But in order to put that regularity 2 theorem to use, we need to start with a L3(D) map. Therefore, we will show ﬁrst that the 2 continuous extension we have just obtained is in L3(D).

The following useful result will allow us to take charts of X where the almost complex structure has a nice expression.

Lemma 4.4.3. Let X be a manifold and J an almost complex structure on X. Let p ∈ X. 2n Then, there exists a chart (U, χ) centered at p, with χ : U −→ χ(U) ⊂ R , such that χ∗(J): R2n −→ End(R2n) (which we also denote by J) satisfy:

(i)J(0) = J0

(ii)DJ(0) = 0 In particular, |J(q) − J(0)| ≤ C|q|2.

Proof. We can always take a chart centered at p in which J(0) = J0. Indeed, it is enough to 2n take a chart centered at p and compose it with a linear isomorphism of R that takes J(0) to J0.

2n 2n 2 Now, observe that if J : R −→ End(R ) is such that J = −Id and J(0) = J0, and ψ : 2n 2n R −→ R is a diﬀeomorphism satisfying ψ(0) = 0 and Dψ(0) = Id, then ψ∗J (which recall is −1 deﬁned by ψ∗J(x) = Dψ(x) J(ψ(x))Dψ(x)) still satisfy ψ∗J(0) = J0. Therefore, it is enough to prove that there exists such a ψ that moreover satisfy D(ψ∗J)(0) = 0. We impose the condition and get:

−1 0 = Dx(ψ∗J(x))|x=0 = Dψ(0) J(ψ(0))(Dx(Dψ)(0)) −1 + Dψ(0) (DxJ(ψ(0))Dψ(0))Dψ(0) −1 −1 − Dψ(0) Dx(Dψ(0))Dψ(0) J(ψ(0))Dψ(0) where we have used that, since Dψ(0)−1Dψ(0) = Id, we have

−1 −1 Dx(Dψ )(0)Dψ(0) + Dψ(0) DxDψ(0) = 0 Taking into account that Dψ(0) = Id and ψ(0) = 0, this simplify to:

0 = J(0)(Dx(Dψ)(0)) + DxJ(0) − Dx(Dψ)(0)J(0)

That is, 0 = [J(0),DxDψ(0)] + DxJ(0).

Now, we must see that we can choose ψ satisfying these equations. Observe that this is a condition only on the second derivatives of ψ at 0. The only restriction we have on the second derivatives is that Dxk Dxl ψ = Dxl Dxk ψ for all k, l. Indeed, given numbers akl satisfying akl = alk, we P 1 P can consider ψ(x) = k xk + 2 k,l aklxkxl. Then, Dψ(0) = Id and Dxk Dxl ψ(0) = akl. The implicit function theorem assures us that ψ is a diﬀeomorphism in some neighbourhood of 0, and this is all we need. 4.4. REMOVAL OF SINGULARITIES 57

We can see that there are numbers Dxk Dxl ψ(0) satisfying 0 = [J(0),DxDψ(0)] + DxJ(0) by seeing that these conditions together with Dxk Dxl ψ(0) is an underdetermined linear system with Dxk Dxl ψ(0) as unknowns. Indeed, we have 4n2 unknowns. Let us count now the number of equations. We have 2n(2n − 1)/2 = 2n2 − n equations from the conditions of the symmetry of the partial derivatives. Let us now count the number of equations coming from [J(0),DxDψ(0)] = −DxJ(0). For eack

1 ≤ k ≤ 2n we have one equation [J(0),Dxk Dψ(0)] = −Dxk J(0). We can write ak bk Dxk J(0) = ck dk 2 Observe that since J = −1 we have Dxk J(0)J(0) + J(0)Dxk J(0) = 0. Recalling that J(0) = J0, this implies that dk = −ak and ck = bk. Therefore, ak bk Dxk J(0) = bk −ak If we write

AB D Dψ(0) = xk CD we see that we can write our equation as

−B − CA − D a b = − k k A − DB + C bk −ak Therefore, we have in fact only n distinct equations for each k. Since we have 2n variables, we get a total of 2n2 equations. Taking into account also the symmetry equations, we have a total of 2n2 + 2n2 − n = 4n2 − n. Hence, we have less equations than unknowns, and the system has a solution. This finishes our proof. The final assertion follows from the Taylor expansion of J around 0, taking into account that DJ(0) = 0. 0 0 ¯ 0 Let u : D −→ X be a continuous map such that u |D∗ is smooth and satisfies ∂J u = 0. From now on, we choose a chart (U, χ) as in the previous lemma centred at the point u0(0), and consider the map u := χ ◦ u. Then, in order to prove the removal of singularities theorem, it will be enough to prove that any J-holomorphic map u : D −→ Cn is smooth at 0.

Observe that the pseudoholomorphic condition ∂J u = 0 can be written as ∂J0 u = g, where g = −(J(u) − J0)∂yu.

2 We can now prove that g ∈ L2(D1/2).

Proposition 4.4.4. Let u : D −→ Cn be a continuous map that is pseudoholomorphic in D∗. 2 Then, g = −(J(u) − J0)∂yu ∈ L2(D1/2). ∗ n ∗ ∗ 1 Proof. Put u0 : D −→ C the restriction of u to D . Then, since D and C = [0, +∞) × S are 0 n 0 −t conformally equivalent, we can see u0 as a map u0 : C −→ C deﬁned by u0(t, θ) = u0(e , θ). By proposition 4.3.4, we have that |∇ku(t, θ)| < Cexp(−σt) for some σ > 1− and t big enough. −t −1 0 −1 σ − Now, observe that (∂iu0)(e , θ) = r (∂iu0)(t, θ). Therefore, |du0(r, θ)| ≤ Cr r ≤ Cr . 2 −1− 3 −2− Similarly, one see that |∇ u0(r, θ)| ≤ Cr and |∇ u0(r, θ) ≤ Cr for r small enough. 58 CHAPTER 4. LOCAL ESTIMATES

Now, we use lemma 4.4.3 to obtain a chart ξ centered at u(0) with the property that |J(p)−J0| ≤ C|p|2, where we call J again to the expression of the almost complex structure J in this chart. Then, consider the expression of u in this local coordinates, that is, ξ ◦ u, which by simplicity will be called also u in what follows.

− Since |du0(r, θ)| ≤ r for r small enough, we have that Z |u(z) − u(0)| ≤ |du(z)| ≤ C|z|1− [0,z] So |u(z)| ≤ C|z|1− for |z| small enough.

1− 2 2−2 If z ∈ D is such that |u(z)| ≤ |z| we have that |J(u(z)) − J0| ≤ C|u(z)| ≤ C|z| . There- − 2−3 fore, since by our previous estimates we have |∂yu(z)| ≤ |z| , we get |g(z)| ≤ C|z| . Since |g|2 is integrable in a neighbourhood of 0, we obtain that g ∈ L2.

We can see similarly that ∇g ∈ L2 and ∇2g ∈ L2.

Indeed,

∇g = ∇J(u)∇(u)∂yu + (J(u) − J0)∇∂yu and

2 2 2 2 ∇ g = ∇ J(u)(∇u) ∂yu + ∇J(u)∇ u∂yu + ∇J(u)∇u∇∂yu 2 + ∇J(u)∇u∇∂yu + (J(u) − J0)∇ ∂yu

2 Taking into account the previous estimates for ||∇u||L2 and ||∇ u||L2 , and the fact that by our choice of chart, we have |∇J(z)| ≤ C|z| and |∇2J(z)| ≤ C0, we obtain the bounds |∇g(z)| ≤ C|z|1−3 and |∇2g(z)| ≤ C|z|−3. Therefore, if we take small enough, we get ∇g ∈ L2 and 2 2 2 ∇ g ∈ L . That is, g ∈ L2.

Lemma 4.4.5. u is a weak solution of ∂J0 u = g in D.

Proof. Observe ﬁrst that the formal adjoint of the operator ∂J0 = ∂x +J0∂y is −∂x +J0∂y = −∂J0

Therefore, in order to check that u is a weak solution of the equation, we only need to check that ∞ R R for any ψ ∈ C (D) we have − D u∂J0 ψ = D gψ.

Fix a small disk of radius > 0 around the origin. Since we know that u is a strong solution of ∗ R R the equation in D , we have that − u∂I ψ = gψ. Therefore, D−D 0 D−D R R R R R R | − u∂I ψ − gψ| = | u∂I ψ + gψ| ≤ |u∂I ψ| + |gψ| D 0 D D 0 D D 0 D

0 But the ﬁrst integral is bounded by |u∂J0 ψ|C A(D) while the second one is as small as we want 2 R R because the integrand is in L (D). Hence, when goes to 0, also does − D u∂J0 ψ = D gψ. Putting together the last two results, we obtain: 4.4. REMOVAL OF SINGULARITIES 59

2 Corollary 4.4.6. u is in L3(D). Proof. The ﬁrst statement follows from proposition 4.4.4, lemma 4.4.5 and the standard elliptic 2 regularity theorem, since ∂J0 u = g, and g ∈ L2. Now we can show that in fact our weak solution u is a strong solution in all of D, using a bootstrapping method. In order to prove this, we will need the following embedding lemma.

2 2 4 2 Lemma 4.4.7. There is an embedding L1([0, 1] ) −→ L ([0, 1] ). In particular, multiplication of 2 2 2 2 2 2 2 2 functions gives a map L1([0, 1] ) × L1([0, 1] ) −→ L ([0, 1] ), that is, if f, g ∈ L1, then fg ∈ L . Proof. This is a special case of a much more general embedding theorem. However, since we only need this special case, and according to our philosophy of using only L2 techniques, we give an elementary proof for this result.

First observe that the second part of the lemma follows from the ﬁrst, since if we have such an R 2 R 4 1/2 R 4 1/2 2 2 embedding, |fg| ≤ ( |f| ) ( |g| ) , so if f, g ∈ L1, then fg ∈ L (where we have used the Cauchy-Schwarz inequality).

Now, let us prove the existence of such an embedding. In fact, we will prove that ||f||L4 ≤ 1/4 R 4 6 ||f|| 2 . For that, it suﬃces to prove that |f| dxdt ≤ 6 for all f : [0, 1] × [0, 1] −→ L1 [0,1]×[0,1] C 2 satisfying f ∈ L1.

Fix such a function f and deﬁne ft : [0, 1] −→ C for any t ∈ [0, 1] by ft(x) = f(t, x). Denoting by < g, h >= gh¯ (the usual Hermitian product), we have:

Z 1 Z 1 ∂ 2 ∂ft(x) |ft(x)| dx = 2Re ft(x), dx ∂t 0 0 ∂t 1/2 Z 1 1/2 Z 1 2 ! 2 ∂ft(x) ≤ 2 |ft(x)| dx dx 0 0 ∂t Z 1 Z 1 2 ! 2 ∂ft(x) ≤ |ft(x)| dx + dx 0 0 ∂t where we have used successively the Cauchy-Schwartz inequality and the Arithmetic-Geometric inequality. Integrating, we get

Z τ 0 Z 1 Z τ 0 Z 1 2 2 ∂ 2 2 ∂ft(x) 2 0 |||fτ |||L2 − ||fτ ||L2 | = |ft(x)| dx ≤ (|ft(x)| + | | )dxdt τ ∂t 0 τ 0 ∂t 2 ≤ ||f|| 2 = 1 L1 for any 0 ≤ τ ≤ τ 0 ≤ 1.

Since

Z 1 2 2 ||ft||L2 dt = ||f||L2 ≤ 1 0 60 CHAPTER 4. LOCAL ESTIMATES

2 we must have ||fτ0 ||L2 ≤ 1 for some τ0 ∈ [0, 1]. Then it follows from the last estimate that 2 ||ft||L2 ≤ 2 for all t ∈ [0, 1].

Now, since the roles of x and t are symmetric, we can interchange them and repeat all the argu- R 1 2 ment to get 0 |ft(x)| dx ≤ 2 for all x ∈ [0, 1].

Deﬁne, for any t, x ∈ [0, 1] Z x 2 Ft(x) := |ft(y)| dy 0 Then, we have

F (x) ≤ ||f ||2 ≤ 2 t t L2 Writing now |f|4 = |f|2|f|2 and using integration by parts in the x variable, we get

Z 1 Z 1 Z 1 Z 1 4 4 |f(t, x)| dxdt = |ft(x)| dx dt 0 0 0 0 Z 1 Z 1 2 2 1 ∂|ft(x)| = |ft| Ft|x=0 − Ft(x)dx dt 0 0 ∂x Z 1 Z 1 2 2 ∂|ft(x)| = |ft(1)| Ft(1) − Ft(x)dx dt 0 0 ∂x Z 1 Z 1 Z 1 2 2 ∂|ft(x)| = |ft(1)| Ft(1)dt − Ft(x)dxdt 0 0 0 ∂x We estimate the two terms separately.

For the ﬁrst term, we have

Z 1 Z 1 2 2 |ft(1)| Ft(1)dt ≤ 2 |ft(1)| dt ≤ 2 · 2 = 4 0 0 And for the second term, using again Cauchy-Schwartz and the Arithmetic-Geometric inequality, we have

Z 1 Z 1 2 Z 1 Z 1 2 ∂|ft(x)| ∂|ft(x)| Ft(x)dxdt ≤ 2 dxdt 0 0 ∂x 0 0 ∂x Z 1 Z 1 ∂ft(x) = 2 2Re ft(x), dxdt 0 0 ∂x 1/2 Z 1 Z 1 1/2 Z 1 Z 1 2 ! 2 ∂ft(x) ≤ 4 |ft(x)| dxdt dxdt 0 0 0 0 ∂x Z 1 Z 1 Z 1 Z 1 2 2 ∂ft(x) ≤ 2 |ft(x)| dxdt + dxdt 0 0 0 0 ∂x 2 = 2||f|| 2 = 2 L1 Putting all together, we ﬁnally obtain 4.4. REMOVAL OF SINGULARITIES 61

4 ||f||L4 ≤ 6 as wanted.

n 2 Proposition 4.4.8. Let u : D −→ C be a map such that u ∈ L3(D), u is a weak solution of n ∗ the equation ∂J0 u = g, and u|D is a pseudoholomorphic curve from the punctured disk into C . Then, u is a pseudoholomorphic curve from the disk into Cn. 2 1 Proof. First of all, observe that u ∈ L3(D) implies that u ∈ C (D).

We will show that u ∈ C∞(D). In order to do that, by the Sobolev lemma, it is enough to prove 2 that u ∈ Lk(D) for all k > 0. Since we know already that u|D∗ is smooth, in order to prove that 2 2 u ∈ Lk(D), it is enough to prove that u ∈ Lk(D) for some > 0.

2 2 We proceed by induction on k. By hypothesis, u ∈ L3(D). Suppose now that u ∈ Lk(D). In 2 2 order to show that u ∈ Lk+1(D) for some > 0, it suﬃces to show that for all h ∈ R (or h equivalently, for all h such that |h| is small enough), ||u || 2 < C for some constant C > 0 Lk h u(x+h)−u(x) independent of h. (Recall that u (x) = |h| ).

By G˚arding’sinequality, we have

h h h ||u || 2 ≤ C(||∂J (u )|| 2 + ||u || 2 ) Lk(D) 0 Lk−1(D) Lk−1 2 h Since u ∈ L (D), we have ||u || 2 ≤ ||u|| 2 , which is independent of h. Let us examine the k Lk−1 Lk h other term, ||∂J (u )|| 2 . 0 Lk−1(D)

Since ∂J0 is a diﬀerential operator with constant coeﬃcients, we have

h h h ∂J0 (u ) = (∂J0 (u)) = ((J(u) − J0)∂xu) Now,

h ((J(u) − J0)∂xu) (x) 1 = ((J(u(x + h)) − J )∂ u(x + h) − (J(u(x)) − J )∂ u(x)) |h| 0 x 0 x 1 = ((J(u(x + h)) − J )∂ u(x + h) − (J(u(x)) − J )∂ u(x + h)) |h| 0 x 0 x 1 + ((J(u(x)) − J )∂ u(x + h) − (J(u(x)) − J )∂ u(x)) |h| 0 x 0 x h h = (J(u) − J0) τh∂xu + (J(u) − J0)(∂xu)

Therefore,

h h h ||∂J (u )|| 2 || ≤ ||(J(u) − J0) τh∂xu|| 2 + ||(J(u) − J0)(∂xu) || 2 0 Lk−1(D) Lk−1(D) Lk−1(D) h h ≤ C||(J(u) − J0) ∂xu|| 2 + ||(J(u) − J0)(∂xu) || 2 Lk−1(D) Lk−1(D) 62 CHAPTER 4. LOCAL ESTIMATES

where in the last step we have used that |τh∂xu(z)| ≤ C|∂xu(z)| (using that the derivative is continuous).

Now we can do a similar analysis as in lemma 4.2.1.

h We start by analysing the base case, k = 3. In this case, we have that ||(J(u) − J0) ∂xu||L2 ≤ h |∂xu|C0 ||(J(u) − J0) ||L2 ≤ C||∇(J(u) − J0)||L2 , where we have used that ∂xu is continuous.

h h Similarly, ||∇((J(u) − J0) )∂xu||L2 and ||(J(u) − J0) ∇(∂xu)||L2 are bounded independently of h, because

h 2 ||∇((J(u) − J0) )∂xu||L2 ≤ |∂xu|C0 ||∇ (J(u) − J0)||L2 and

h ||(J(u) − J ) ∇(∂ u)|| 2 ≤ C||∇(J(u) − J )∇(∂ u)|| 2 ≤ |∇(J(u) − J )| 0 ||∂ u|| 2 0 x L 0 x L 0 C x L1

2 h h 2 h Finally, ||∇ (J(u) − J0) ∂xu||L2 , ||(J(u) − J0) ∇ (∂xu)||L2 and ||∇(J(u) − J0) )∇(∂xu)||L2 are bounded independently of h, the ﬁrst two with similar arguments as above (using that 2 2 ∂xu ∈ L2 and J(u) − J0 ∈ L2, hence they are continuous), and the last one, using that h 2 ||∇(J(u) − J0) )∇(∂xu)||L2 ≤ C||∇ (J(u) − J0)∇(∂xu)||L2 , and applying lemma 4.4.7 taking 2 2 into account that ∇ (J(u) − J0), ∇(∂xu) ∈ L1.

h Therefore, ||(J(u) − J )(∂ u) || 2 is bounded independently of h. 0 x L2

Now, the same kind of analysis allows us to write

h h ||(J(u) − J )(∂ u) || 2 ≤ C + |J(u) − J | 0 ||u || 2 0 x L2 0 C L3 p q h 2 Indeed, we can bound all the terms ||∇ (J(u)−J0)∇ ((∂xu) )||L for p+q ≤ 2 as before, except for h h the term p = 0, q = 2, for which we get the bound |J(u)−J | 0 ||∂ (u )|| 2 ≤ |J(u)−J | 0 ||u || 2 . 0 C x L2 0 C L3 Summing up, we have:

h 0 h ||u || 2 ≤ C + C |J(u) − J | 0 ||u || 2 L3 0 C L3 0 Taking the radius of the disk, , small enough, we can ensure that C |J(u) − J0|C0 < 1/2. Then, we get:

h ||u || 2 ≤ 2C L3(D) h 2 Hence, since ||u || 2 is bounded independently of h, we obtain that u ∈ L (D ). L3(D) 4

We can now consider the general case, k > 3. We proceed in the same way as before, only that in this case we don’t need to use lemma 4.4.7. The analogous analysis as in the previous case 2 k−2 k−2 (taking into account that if u ∈ Lk then u ∈ C and I(u) − I0 ∈ C ) allows us to conclude 2 that u ∈ Lk+1(D), for a suitable .

2 Therefore, we obtain that in fact u ∈ Lk(D1/2) for all k, and by the Sobolev embedding theorem, u ∈ C∞(D) (since our argument shows that it is smooth at the origin, and it is smooth outside ∞ the origin by hypothesis). That is, u is a C solution to ∂J u = 0 in D. 4.4. REMOVAL OF SINGULARITIES 63

With this, we ﬁnish the proof of the removal of singularities theorem. 64 CHAPTER 4. LOCAL ESTIMATES Chapter 5

Proof of the Gromov compactness theorem

In this chapter we finish the proof of the Gromov compactness theorem, using the results obtained in the last chapter. In the first section we analyse how bubbling occurs and prove a first rough compactness theorem for pseudoholomorphic curves. In the second section we study in more depth the phenomenon of bubbling, and in the final section we put all together and interpret bubbling in terms of Gromov convergence, giving a proof of the Gromov compactness theorem.

In this chapter we follow closely [M-S1].

5.1 Bubbling

In this section we will see how bubbling appears as a consequence of the conformal invariance of the energy. Recall that we have seen a manifestation of this phenomenon in section 2.3. Moreover, as a ﬁrst step towards the Gromov compactness theorem, we will prove the following compactness theorem.

Proposition 5.1.1. Let Jn be ω-compatible almost complex structures on X converging in the ∞ 2 C sense to an ω-compatible almost complex structure J. Let un : S −→ X be a sequence of Jn-holomorphic curves with supn E(un) < ∞. Then there exists a ﬁnite set of points F = 1 n 2 {z , ..., z } ⊂ S , a subsequence of un (which we again denote by un for simplicity) and a J- holomorphic curve u : S2 −→ X such that:

2 1. un converge uniformly to u with all derivatives in compact subsets of S − F . j j 2. For every j and every > 0 such that D(z ) ∩ F = {z }, the limit

j m(z ) := lim E(un; D(zj)) n exists and is a continuous function of , and

j j m(z ) := lim m(z ) ≥ →0 }

65 66 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

3. For every compact subset K ⊂ S2 with F ⊂ ◦K,

l X j E(u; K) + m(z ) = lim E(un; K) n j=1

First of all, we prove the following important proposition, telling us that a non-constant pseudoholomorphic sphere has a lower bound to its energy depending only on the target manifold.

Proposition 5.1.2. There exists a constant } > 0, depending only on the symplectic manifold (X, ω) and the ω-compatible almost complex structure J, such that if u : S2 −→ X is a non- constant pseudoholomorphic map, then E(u) ≥ }.

Moreover, if we denote }(J) the constant corresponding to the almost complex structure J, and ∞ Jn is a sequence of ω-compatible almost complex structures converging to J in the C sense, then }(Jn) converges to }(J). Proof. Assume that the proposition is false. Then, for each j ∈ N there exists a non-constant 2 1 pseudoholomorphic map uj : S −→ X such that E(uj) < j .

2 Let > 0. Cover S by ﬁnitely many open charts with chart maps ξi : Ui −→ D1 such that −1 2 −1 0 the sets ξi (D1/2) also cover S . Then, taking j large enough we have that |d(uj ◦ ξi )|C (D1/2) is as small as we want for all i, by corollary 4.2.5. Therefore, we have |duj|C0 < for j large 2 enough. In this case, we have that diam(uj(S )) < 2π. If we take small enough so that any set with diameter less than 2π is contained in a coordinate chart of M homeomorphic to the open ball of radius 1, B1 (this always exists since X is compact, so an open covering with co- 2 ordinate charts homeomorphic to B1 has a Lebesgue number), then uj(S ) is contained in such a coordinate chart of X. Since these coordinate charts of X are contractible, we have that uj is homotopic to a constant map. Therefore, E(uj) = 0 (because energy is a topological invariant of a pseudoholomorphic curve), which is a contradiction with the fact that uj is not constant.

For the second part, observe that the only place where the almost complex structure enters is in the deﬁnition of necessary to apply corollary 4.2.5. But this number depends continuously on the almost complex structure J, since by looking at the proof of corollary 4.2.5, we see that it is used in order to apply proposition 4.2.3, which can be applied to any sequence of Jn-holomorphic ∞ maps, where Jn converge to J in the C sense. Observe that in this proposition we make full use of the condition that J is ω-compatible, since this is what allows us to say that the energy depends only on the class of homology that the curve represents. Because of this proposition, one often says that the energy of a pseudoholomorphic sphere is quantized.

Now we can give a proof of proposition 5.1.1.

2 Suppose that un is a sequence of Jn-holomorphic curves un : S −→ X such that supn E(un) < ∞. If we have supn |dun|C0(S2) < ∞, we can apply proposition 4.2.3 and conclude that a subsequence converges in all of S2. Therefore the only failure of compactness occur when we have 2 supn |dun|C0(S2) = ∞. Let zn ∈ S be such that cn := |dun(zn)| = |dun|C0(S2). Passing to a 2 subsequence we can assume that zn converges to a point z0 ∈ S and cn diverges to inﬁnity. 5.1. BUBBLING 67

2 −1 Consider a map φ : U −→ φ(U) ⊂ S where U ⊂ C, φ(0) = z0 and φ is a complex chart 2 1 in S such that φ(0) = z0. We can assume without loss of generality that 2 ≤ |dφ(z)| ≤ 2 for all z ∈ U and |dφ(0)| = 1. Moreover, we can also assume without loss of generality (maybe 0 0 −1 passing to a subsequence) that zn ∈ φ(U) for all n. Put un = un ◦ φ and zn = φ (zn). Then, 0 0 0 1 0 0 0 |dun(zn)| = |du(zn)||dφ(zn)|, and therefore 2 cn ≤ |dun(zn)| ≤ cn and limn zn = 0.

Now we pick > 0 such that D(zn) ⊂ U and consider the reparametrized sequence vn : 0 0 1 1 Dc −→ X deﬁned as vn(z) = u (zn + z/cn). Then, |dvn(0)| = |du (zn)| ≥ , and n n n cn 2 1 0 0 |dvn| 0 ≤ |dun|C ≤ 2. Moreover, we have E(vn; Dc ) = E(u ; D(zn)) ≤ E(u ) ≤ E(un), C 0 cn n n n where we have used the conformal invariance of energy twice, since both φ and z 7→ zn + z/cn are conformal maps.

We can now apply proposition 4.2.3 to the sequence vn (since we have seen that it has uniformly bounded derivative) to get a J-holomorphic curve v : C −→ X. Moreover, by the previous bounds 1 we have |dv(0)| ≥ 2 (in particular, v is not constant, so E(v) > 0), and E(v) ≤ supi E(ui) < ∞, so v has finite energy. Consider the curvev ˜ : C − {0} −→ X defined byv ˜(z) = v(1/z). By the conformal invariance of energy,v ˜ has finite energy. Therefore, by the removal of singularities theorem,v ˜ extends to a J-holomorphic curve defined in the whole C. Therefore, we can see v as a J-holomorphic curve v : S2 −→ X. Moreover, as we have noted above, v is not constant, so by proposition 5.1.2 we have that E(v) > }. Observe also that for any R > 0 and every > 0 we have

0 0 0 0 E(v; DR) = lim E(vn; DR) = lim E(un; DR/cn (zn)) ≤ lim E(un; D(zn)) n n n where in the last step we have used that cn diverges to inﬁnity. Taking the limit R → ∞, 0 0 we ﬁnally obtain E(v) ≤ lim infn E(un; D(zn)) for every > 0. Therefore, also for any > 0 we have } ≤ E(v) ≤ lim infn E(un; D(zn)), where we have used that φ is conformal, 0 0 so E(un; D(zn)) = E(un; D(zn)).

The maps v : S2 −→ X arising in this way are called bubbles, and it is said that v bubbles off in the limit. Observe that we have just seen that any sequence of points in S2 where the derivatives dun diverge gives rise to a non constant bubble in the limit. Moreover, the energy E(v) comes from arbitrarily small neighbourhoods of the points zn after a suitable conformal rescaling of the sequence. Since any bubble has energy at least } > 0 and the energy of the sequence is bounded, this implies that there can only be a finite number of bubbles that bubble off in the limit.

Let us formalize this argument in order to have a proper proof of proposition 5.1.1.

Proof of Proposition 5.1.1. We start with a sequence un of Jn-holomorphic curves with bounded energy, that is, supn E(un) < ∞. Suppose supn |dun|C0 = ∞. Then we can apply the previous 1 2 construction and passing to a subsequence un find a sequence of points zn of S converging to 2 z ∈ S and such that |dun(zn)| → ∞, so that a bubble bubbles off at z in the limit. Now, if it 2 1 1 1 2 1 exists, we can take a subsequence un and points zn such that zn → z 6= z and |dun(zn)| → ∞, so that by the previous argument a bubble bubbles off at z1 in the limit. We continue this process until it stops.

Now we will show that this process must necessarily stop in a ﬁnite number of steps. In- supn E(un) N deed, it must stop in at most steps. For suppose we have a subsequence (u )n } n 1 N 2 supn E(un) j 2 and points z , ..., z ∈ S with N > such that there are sequences (z )n ⊂ S for } n 68 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

j j N j j = 1, ..., N satisfying zn → z and |dun (zn)| → ∞ for each j = 1, ..., N. Choose > 0 1 N such that the balls D(z ), ..., D(z ) are all disjoint. By the above estimates we have that j } ≤ lim infn E(un; D(z )) for every j = 1, ..., N. So, since the balls are disjoint,

N X j sup E(un) ≥ lim inf E(un) ≥ lim inf E(un; D(z )) ≥ N} > sup E(un) n n n n j=1 a contradiction.

1 N N Put F = {z , ..., z }, and for simplicity call un to the subsequence un obtained at the end of ◦ the process. Consider a countable family of compact sets Ki such that Ki ⊂ Ki+1 ⊂ Ki+1 for 2 S 0 each i and such that i Ki = S − F . By construction we have |dun|C (Ki) < ∞ for each i. Therefore, we can apply proposition 4.2.3 and conclude that there exists a subsequence of the i ∞ un that converges to a J-holomorphic curve u : Ki −→ X in the C sense in Ki for each i. Finally, by using a diagonal argument we obtain a subsequence of the un that converges to a J-holomorphic curve u∞ : S2 − F −→ X in the C∞ sense (that is, it converges uniformly in ∞ compact with all its derivatives). Now just note that E(u ) < supn E(un) < ∞, so that it has ﬁnite energy and then by N applications of the removal of singularities theorem (at each of the 2 points z1, ..., zN ) we obtain a pseudoholomorphic curve u : S −→ X satisfying the condition (i).

j j Now we take a further subsequence such that m(z ) = limn E(un; D(z )) exist for all j. This subsequence fulfills all the conditions of the proposition. It is clear that it still satisfies (i). For 0 j j (ii), take 0 < < . Then, since ui converge uniformly in the annulus D(z ) − D0 (z ), we j j j j j 0 have that m(z ) = m0 (z ) + E(u; D(z ) − D0 (z )), so clearly m0 (z ) is defined for all < 0 j and is continous in . Since as we have seen before, we have m(z ) ≥ } for all > 0, we have m(zj) ≥ } for all j.

j Finally, for (iii), since the D(z ) are disjoint we have that

n n n 2 [ j X j X j E(u; S − D(z )) = lim E(ui; K) − lim E(ui; D(z )) = lim E(ui; K) − m(z ) i i i j=1 j=1 j=1 and taking the limit when → 0, we obtain

n 2 X j E(u; S − F ) = lim E(ui; K) − m(z ) i j=1

Observe that we have now a compactness theorem, but we have not said anything about the behaviour of the limit at the singular points in F . By the above argument we know that under a suitable rescaling of the sequence a bubble appears at each of this points. But this is not the end of the story, since multiple bubbling or nested bubbling can appear at a given point, so that if we just consider the bubble given by the argument above (which is a single bubble) we are losing energy in the process. This happens because we have not taken into account the behaviour of the sequence in small annuli around the singular points. In the next section we take a closer look at this and use the results obtained to give a full proof of the Gromov compactness theorem. 5.2. SOFT RESCALING 69

5.2 Soft rescaling

In this section we will improve the results we have just obtained by analyzing in detail the way in which bubbling occurs. In particular, we will see that by reﬁning our bubbling argument, choosing apropriately the rescaling of the un’s, we can obtain a bubble that connects to the limit of un obtained in proposition 5.1.1. In order to prove this reﬁned compactness theorem, we will have to study carefully the behaviour of the sequence in annuli around a singular point. This annuli can be thought as the necks that join the principal bubble with a developing bubble, so that in the limit the necks contract to a single point and the resulting limit is singular.

The main result of this section is the following:

Theorem 5.2.1. Let Jn be ω-compatible almost complex structures on X that converge in the ∞ C sense to an ω-compatible almost complex structure J. Let z0 ∈ C and r > 0. Let un : Dr(z0) −→ X be a sequence of Jn-holomorphic curves and u : Dr(z0) −→ X a J-holomorphic map such that the following holds

∞ (a) un converges to u in the C sense on Dr(z0) − {z0}.

(b) m0 := lim lim E(un; D(z0)) exists and m0 > 0. →0 n

Then, there exist a subsequence (which we denote also by un for simplicity), a sequence of Moebius 2 transformations ψn ∈ G, a J-holomorphic sphere v : S −→ X and ﬁnitely many distinct points 2 z1, ..., zl, z∞ ∈ S such that the following holds

∞ 2 (i) ψn converges to z0 in the C sense on compact subsets of S − {z∞}.

∞ (ii) The sequence vn := un ◦ ψn converges to v in the C sense on all compact subsets of 2 S − {z1, ..., zl, z∞}, and the limits

mj := lim lim E(vn; D(zj)) →0 n

exist and satisfy mj > 0 for all j = 1, ..., l. Pl (iii) E(v) + j=1 mj = m0 (iv) If v is constant, then l ≥ 2.

Proof. Since the proof is quite long, we will split it in 4 steps.

Step 1: We may assume without loss of generality that z0 = 0 and |dun(z)| reaches its maximum at z = 0, for all n.

Since by (a), u and its derivative converge uniformly away from z0, the sequence |dun| is uniformly bounded in the annulus Dr(z0) − D(z0) for every > 0. We have also, for every > 0, that

lim sup |dun| = ∞ n D(z0) R 2 for otherwise, we would had E(un; D(z0)) = |dun| ≤ C for some 0 < C < ∞ indepen- D(z0) dent of n, and hence m0 ≤ lim→0 C = 0, a contradiction with (b). 70 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

Therefore, every sequence z ∈ D (z ) with |du (z )| = sup |du | converges to z (oth- n R 0 n n DR(z0) n 0 erwise, the sequence stays inﬁnitely often in some annulus Dr(z0) − D(z0) where the energy density is bounded, contradicting the hypothesis that the energy density reaches its maximum at the points of the sequence). We may then assume zn ∈ Dr/2(z0) for all i (by passing to a subsequence of the un).

Consider the mapsu ñ : DR/2 −→ X defined byu ñ(z) := un(z + zn) (observe that the map z 7→ z + zn is in G). These new maps clearly satisfy the hypothesis (a) and (b), with r replaced by r/2 and z0 replaced by 0. Moreover, if they satisfy the conclusion for some sequence ψn ∈ G, so do the old maps with the sequence z 7→ ψn(z) + zn. This finishes step 1.

Step 2: Deﬁnition of the sequence ψn ∈ G satisfying condition (ii) and condition (i) with z0 = 0 and z∞ = ∞.

Choose 0 < δ < } be such that lemma 4.3.5 holds with the of the lemma equal to δ. By passing to a subsequence, we may choose a sequence of numbers δn > 0 such that

δ E(u ; D ) = m − n δn 0 2 Let us see that such a choice (for n large enough) is possible. Observe that, by passing to a subsequence, for a ﬁxed n, we have that E(un; D) is strictly positive and decreases continuously with → 0 with limit 0. On the other hand, for a ﬁxed > 0, we have that m = limn E(un; D) ≥ m0 δ and is decreasing with . Therefore, for n large enough, we have E(un; D) > m0 − 2 and it δ follows that there is some δn with E(un; Dδn ) = m0 − 2 .

We see now that δn → 0. Indeed, suppose this is not the case. Then there is an 0 > 0 such that for any n0 there exists n > n0 such that δn > 0. So, passing to a subsequence we may assume that δn > 0 for all n. But this is a contradiction, since we would have:

m0 = lim lim E(un; D) ≤ lim E(un; Dδ ) < m0 →0 n n n

Let us explain intuitively what this condition means. When we take Dδn , we are taking neighbourhoods of z0 such that all bubbling near z0 occur inside this neighbourhoods. This is because m0 is the energy of all the bubbles occurring at z0 in the limit, while outside this neighbourhood we have energy at most m0 − }/2 and any non-constant bubble has energy at least }. Therefore, by considering these neighbourhoods we are considering all the zone near z0 relevant to bubbling.

Consider now the sequences ψn ∈ G and vn : DR/δn −→ X deﬁned by ψn(z) := δnz and vn(z) := un(δiz). Since δn → 0, we have that ψn satisﬁes (i) with z∞ = ∞. Moreover,

|dvn(0)| = sup |dvn| (5.1) Dr if r < R/δn (recall that |dun(z)| reaches its maximum at z = 0). The energy of vn in Dr is bounded by the energy of un in DR, for any ﬁxed r and n large enough. Then, we can apply 2 proposition 5.1.1 to the vn and obtain a subsequence (again called vn) and points z1, ..., zl ∈ S 2 such that (ii) holds. Let F = {z1, ..., zl} ⊂ S . Suppose that F 6= ∅. Then, equation 5.1 implies that 0 ∈ F , by our discussion on bubbling in the previous section. This ﬁnishes the proof of step 2. 5.2. SOFT RESCALING 71

Now we have already obtained the desired subsequence of the un and the desired maps ψn. It only remains to show that conditions (iii) ad (iv) are fulﬁlled. In order to do that, we will need the following identity.

Step 3: lim lim E(un; DRδ ) = m0 R→∞ n n The proof is by contradiction. We assume that it is false. Then, there exists ρ > 0 and a subsequence (also called un) such that for every R ≥ 1,

lim E(un; DRδ ) ≤ m0 − ρ n n Indeed, this follows from the fact that, for a ﬁxed R ≥ 1, we have

lim E(un; DRδ ) ≤ m0 n n because for any > 0 there is an n0 such that Rδn < , hence for all n ≥ n0, E(un, DRδn ) ≤

E(un; D), so limn E(un; DRδn ) ≤ limn E(un; D), and therefore limn E(un; DRδn ) ≤ lim→0 limn E(un; D) = m0.

So, using the deﬁnition of δn, we have that for every R ≥ 1, δ lim E(un; A(δn, Rδn)) = lim E(un; DRδ ) − lim E(un; Dδ ) ≤ − ρ (5.2) n n n n n 2

Now we show that there is a sequence of numbers n > 0 such that limn n = 0 and limn E(un; Dn ) = m0. Indeed, by the deﬁnition of m0, for each l ∈ N there exists l ∈ (0, 1/l) and nl ∈ N such that |E(un; l) − m0| ≤ 1/l for all n > nl. Without loss of generality we can assume also that l+1 < l and nl+1 > nl for all l. Then, we can deﬁne n := l for nl ≤ n < nl+1.

Since Rn → 0 for each ﬁxed R ≥ 1, we also have for R ≥ 1

lim E(un; DR ) = m0 (5.3) n n where we use that, as with the δn, we have limn E(un; DRn ) ≤ m0 and limn E(un; Dn ) ≤ limn E(un; DRn ).

Observe that the ratios n/δn are not bounded on any subsequence. Because otherwise, by passing to a subsequence we get n < Cδn for some C ≥ 1 and that would imply m0 = limn E(un; Dn ) ≤ limn E(un; DCδn ) ≤ m0 − ρ, a contradiction.

So, by the deﬁnition of n and δn, we have

lim E(un; A(δn, n)) = δ/2 (5.4) n and

δ lim n = 0 n n

Therefore, for n large enough we have δn < n and we can consider the annuli A(δn, n). Now we will examine the behaviour of the sequence on these annuli. For each T > 0 (such that it makes 72 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

T T −T −T sense) we split each of these annuli in three annuli A(δn, e δn), A(e δn, e n) and A(e n, n).

We will prove

T lim lim E(un,A(δn, e δn)) = δ/2 (5.5) T →∞ n which leads to a contradiction with equation 5.2 and establishes step 3.

To prove it, consider the sequence wn : D1/n −→ X deﬁned by wn(z) := un(nz). By passing ∞ to a subsequence, we may assume that wn converges in the C sense to a pseudoholomorphic map w outside a ﬁnite set of points. Using that δn/n → 0, we have for each T ≥ 1 that

−T T −T T T } lim E(wn; A(e , e )) = lim E(un; A(e n, e n)) ≤ lim E(un; A(δn, e n)) ≤ n n n 2 where in the last inequality we have used equation 5.3 and the deﬁnition of δn. This implies that ∞ wn converges to a constant in the C sense on C − {0}, because the bound imply that there ∞ is no bubbling there, hence for any compact K ⊂ C − {0} wn converges to w in the C sense 0 0 0 on K and in particular |dw|C (K) = limn |dwn|C (K) = limn n|dun|C (K/n) = 0, where we have ∞ used that un converges in the C sense away from 0.

Therefore, for all T ≥ 0,

−T −T −T lim E(un; A(e n, n)) = lim E(wn; A(e , 1)) = E(w; A(e , 1)) = 0 n n Now, by equation 5.4, the deﬁnition of δ and lemma 4.3.5 with σ = 1/2, we have that there exists a constant C > 0 such that

T −T −T δ lim E(un; A(e δn, e n)) ≤ Ce n 2 −T Since limn E(un; A(δn, n)) = δ/2 and limn E(un; A(e n, n)) = 0, we have that

T −T δ lim E(un; A(δn, e δn)) ≥ (1 − Ce ) n 2 T Taking the limit when T → ∞, and taking into account that equation 5.4 shows that limn E(un; A(δn, e δn)) ≤ δ 2 , this proves 5.5, and thus establishes step 3.

Step 4: Finishing the proof.

Let us prove (iii). Since limR→∞ limn E(un; DRδn ) = m0, and recalling the deﬁnition of δn, we have that

lim lim E(vn; DR) = lim lim E(un; DRδ ) = m0 R→∞ n R→∞ n n and for every n,

δ E(v ; D ) = E(u ; D ) = m − ≥ m − } n 1 n δn 0 2 0 2

Therefore, all the bubble points z1, ..., zl of the sequence vi lie in the disk D1. Fix a number s > 1. Then, we have 5.2. SOFT RESCALING 73

m0 = lim lim E(vn; DR) R→∞ n

= lim lim E(vn; DR − Ds) + lim lim E(vn; Ds) R→∞ n R→∞ n

= lim E(v; DR − Ds) + lim E(vn; Ds) R→∞ n  l  l [ X = E(v; − Ds) + lim lim E vn; Ds − D(zj) + mj C →0 n j=1 j=1 l X = E(v) + mj j=1 where we have used the identity of step 3. This proves (iii).

Let us prove (iv). Since all bubble points of vn lie in the closed unit disk, if v is constant and 1 < R1 < R2, we have

lim E(vn; A(R1,R2)) = E(v; A(R1,R2)) = 0 n

Therefore the limit limn E(vn; DR) = limn E(un; DRδn ) is independent of R > 1. So by the δ identity of step 3, limn E(vn; DR) = m0 for every R > 1. Since E(vn; D1) = m0 − 2 , we have limn E(vn; D1) < m0. This is only possible if there is a bubble point z of the sequence vn lying on the boundary of the unit disk (that is, with |z| = 1). But we have seen at step 2 that if the set of singular points F for v is not empty, then also 0 ∈ F , so l = #F ≥ 2, as wanted.

This finishes the proof. Now we prove a lemma telling us that if when a bubble bubbles off in the limit there is no loss of energy, then the bubble connects with the principal component, in the sense that the point where the bubble develops in the principal component and that of the bubble have the same image. We will prove this first in a special situation (the following lemma) and then proving that the general situation can be reduced to the situation of the lemma. This, together with the last result, makes plausible that the limit of a sequence of pseudoholomorphic maps converge to a stable map. We will prove this in the next section.

Lemma 5.2.2. Let Jn be a sequence of ω-compatible almost complex structures converging in the C∞ sense to an ω-compatible almost complex structure J. Let r > 0 and a positive sequence δn → 0. Let un : A(δn/r, r) −→ X be a sequence of J − n-holomorphic curves satisfying

∞ (i) un converge to a J-holomorphic curve u : Dr −→ X in the C sense on Dr − {0}. 2 ∞ (ii) un(δn·) converge to a J-holomorphic curve v : S − D1/r −→ X in the C sense on 2 C − D1/r (where here we think of C as a subset of S ).

(iii) lim lim E(un; A(δn/ρ, ρ)) = 0 ρ→0 n Then, we have

u(0) = v(∞) 74 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

Proof. Observe that the annuli A(r, R) is conformally equivalent to a ﬁnite cylinder, so the total energy remains invariant. Fix δ as the in lemma 4.3.5. By (iii), we have that there exists ρ > 0 (which we take such that 2ρ < r) and n0 such that for all n > n0 we have E(un; A(δn/(2ρ), 2ρ)) < δ. Then by lemma 4.3.5 with σ = 1/2 and T = log 2 we obtain that T −T E(un; A(δn/ρe , e ρ) ≤ 2CE(un). Now, by using corollaries 4.2.5 and 4.2.2 applied to obtain bounds over all the annulus (using the standard argument of covering it by suitable small open sets, applying there the corollaries and using the compactness of the annulus), we get

0 00 0 2 2 |dun|C (A(δn/ρ,ρ)) ≤ C ||dun||L (A(δn/ρ,ρ)) ≤ C ||dun||L (A(δn/(2ρ),2ρ)) and Z 0 0 2 d(un(z), un(z )) ≤ |dun| ≤ r|dun|C (A(δn/ρ,ρ)) ≤ C||dun||L (A(δn/(2ρ),2ρ)) [z,z0] 0 for z, z ∈ A(δn/ρ, ρ).

From this, taking the limit when n → ∞

d(u(ρ), v(1/ρ)) = lim d(un(ρ), un(δn/ρ)) ≤ C lim ||dun||L2(A(δ /(2ρ),2ρ)) n n n p 2 To ﬁnsh, just observe that ||dun||L (A(δn/(2ρ),2ρ)) = E(un; A(δn/(2ρ), 2ρ)) and that, by (iii), limρ→0 limn E(un; A(δn/(2ρ), 2ρ)) = 0, so taking the limit ρ → 0, we get

d(u(0), v(∞)) = lim d(u(ρ), v(1/ρ)) ≤ lim lim E(un; A(δn/(2ρ), 2ρ)) = 0 ρ→0 ρ→0 n That is, u(0) = v(∞), as wanted.

Proposition 5.2.3. Let Jn be a sequence of ω-compatible almost complex structures converg- ∞ ing in the C sense to an ω-compatible almost complex structure J. Let z0 ∈ C and u, un : Dr(z0) −→ X a J-holomorphic and Jn-holomorphic (respectively) curves satisfying the following conditions

∞ (a) un converges to u in the C sense on Dr(z0) − {z0}

(b) The limit m0 := lim lim E(un; D(z0)) exists and m0 > 0. →0 n

2 2 Suppose moreover that ψn ∈ G, v : S −→ X and z1, ..., zn, z∞ ∈ S satisfy

∞ 2 (i) ψn converges in the C sense to z0 on every compact subset of S − {z∞}.

∞ 2 (ii) The sequence vn := un ◦ ψn converges to v in the C sense on S − {z1, ..., zl, z∞} and the limits

mj := lim lim E(vn; D(zj)) →0 n

exist and satisfy mj > 0 for all j = 1, ..., l. Pl (iii) E(v) + j=1 mj = m0 5.2. SOFT RESCALING 75

Then, u(z0) = v(z∞).

Proof. We will reduce this situation to the particular case of the previous lemma.

First of all, we will show that we can assume without loss of generality that z0 = 0 and z∞ = ∞. For this, choose isometries φ0, φ∞ ∈ G such that φ0(z0) = 0 and φ∞(z∞) = ∞. Now, replacing −1 −1 ˜ −1 −1 un, vn, ψn, u and v withu ˜n = un ◦ φ0 ,v ˜n = vn ◦ φ∞ , ψn = φ0 ◦ ψn ◦ φ∞ ,u ˜ = u ◦ φ0 and −1 v˜ = v ◦ φ∞ respectively, we have that they still satisfy the conditions of the statement (we use that elements of G preserve the energy of the curve and that the group of isometries of S2 is compact, so composing with φ0, φ∞ does not spoil convergence in condition (i)) and they satisfy −1 −1 u˜(0) = u(φ0 (0)) = u(z0) = v(z∞) = v(φ∞ (∞)) =v ˜(∞). Moreover, it is clear that if the new maps satisfy the conclusion of the proposition (˜u(0) =v ˜(∞)) then also the old maps satisfy it.

Now we show that again without loss of generality we can assume that the maps ψn are of the form ψn(z) = δnz, where δi → 0. Observe that by hypothesis we have that ψn(0) → 0 −1 and ψn (∞) → ∞. The last assertion follows because otherwise there exists a subsequence 2 2 ψj and a sequence of points zj ∈ S with zj ∈ K ⊂ S − {∞} (K compact) for all j such that ψj(zj) = ∞, contradicting the fact that ψn converges to 0 uniformly on compact sets of 2 0 ∞ S − {∞}. Then, we can find sequences ρn, ρn ∈ G that converge uniformly to the identity 0 0 ∞ −1 ∞ such that ρn(ψn(0)) = 0, ρn(∞) = ∞, ρn (ψn (∞)) = ∞, ρn (0) = 0, and for n large enough, ∞ 0 ρn (1) = 1 and ρn(ψn(1)) =: δn ∈ R+. Indeed, recall that an element of G is completely determined by its image in three distinct points of S2 and that G acts transitively in the set of 2 0 ∞ all three distinct points of S , so ρn and ρn are completely determined by this conditions, and 2 they converge to the identity, since in the limit they fix three points of S . Now replace un, vn 0 −1 ∞ −1 ˜ 0 ∞ −1 and ψn byu ñ := un ◦ (ρi ) ,v ñ := vn ◦ (ρn ) and ψn := ρn ◦ ψn ◦ (ρn ) . Then, we have ˜ 0 ˜ 0 ˜ 0 ψn(1) = ρn(ψn(1)) = δn, ψn(0) = ρn(ψn(0)) = 0 = δn0 and ψn(∞) = ρn(∞) = ∞ = δn∞, so ˜ 2 ψn(z) = δnz for all z ∈ S (since both are elements of G and coincide in three points). Moreover, if the conclusion of the theorem holds for the new maps, then it holds for the old maps also.

We want to prove now that E(ρ) := limn E(un; A(δn/ρ, ρ)) converges to 0 as ρ → 0. Indeed, by using the conformal invariance of energy, we obtain

E(ρ) = lim E(un; Dρ) − lim E(vn; D1/ρ) n n l X = m0 + E(u; Dρ) − E(v; D1/ρ) − mj j=1

= E(u; Dρ) + E(v; C − D1/ρ)

where in the second equality we have used that limn E(un; Dρ) = E(u; Dρ)+m0 (by deﬁnition of Pl m0) and limn E(vn; Dρ) = E(v; Dρ) + j=1 mj (by deﬁnition of mj), and in the third equality we have condition (iii) from the statement.

To ﬁnish, note that we have just proved that we are in the conditions of the previous lemma. So its application gives us u(0) = v(∞), as wanted. 76 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

5.3 Gromov compactness

Now that we have all the hard work done, we will put everything together and interpret the obtained results in terms of the notion of Gromov convergence.

Theorem 5.3.1 (Gromov compactness). Let Jn be ω-compatible almost complex structures con- ∞ 2 verging, in the C sense, to an ω-compatible almost complex structure J. Let un : S −→ X be a sequence of Jn-holomorphic curves such that supn E(ui) < ∞. Then, un has a Gromov convergent subsequence.

Proof. Without loss of generality (by passing to a subsequence) we may assume that E(un) converges. We put

E = lim E(un) n Before we actually start the proof, we will introduce some useful notation. We will describe a tree T with N vertices as a vector j = (j2, ..., jN ) with jk < k for all k. Let us explain how to do it. We can enumerate the vertices of the tree v1, ..., vN in such a way that Ti := {v1, ..vi} is a tree for any i. Indeed, start with any vertex v1 an put v2, ..., vi1 a maximal path starting at v1.

If there are no more vertices in T we are done. Otherwise, choose a vertex vk ∈ {v1, v2, ..., vi1 that is related to a vertex not yet considered, and choose again a maximal path starting with vk. We continue in this way until we have all the vertices of T (the process stops because T is ﬁnite). It is clear that with the vertices chosen in this way, Ti is a tree for all i. Observe that by deﬁnition of the ordering, for each i there is some ji < i such that viEvji . Therefore, the vector j = (j2, ..., jN ) with jk < k for all k determines completely the tree T , since it encodes the relation E.

We will construct by induction

1 2 N 2 (a) a (2N − 2)-tuple u = (u , u , ..., u ; j2, ..., jN ; z2, ..., zN ), where un : S −→ X are pseudoholomorphic spheres, jk < k for k ≥ 2 are positive integers, and zk ∈ C with |zk| ≤ 1.

(b) ﬁnite subsets Fk ⊂ D1 for k = 1, ..., N. k (c) sequences of Moebius transformations {φn}n∈N for k = 1, ..., N such that some subsequence satisﬁes the following conditions

1 1 ∞ 2 k (i) un ◦ φn converges to u in the C sense on S − F1. For k = 2, ..., N, un ◦ φn converges to k ∞ u in the C sense on C − Fk. Moreover, F1 ⊂ {0} and ZN = ∅. k (ii) If u is constant, then #Fk ≥ 2. (iii) The limit

k mk(z) := lim lim E(un ◦ φn; D(z)) →0 n

1 exists and satisﬁes mk(z) > 0 for all z ∈ Fk. If F1 = {0} then E = E(u ) + m1(0). If k ≥ 2

then zk ∈ Fjk and

k X mjk (zk) = E(u ) + mk(z)

z∈Zk 5.3. GROMOV COMPACTNESS 77

(iv) If jk = jk0 then zk 6= zk0 .

j k (v) For k = 2, ..., N, u k (zk) = u (∞)

jk −1 k ∞ 2 (vi) For k = 2, ..., N,(φn ) ◦ φn converges to zk in the C sense on C = S − {∞}.

(vii) For k = 2, ..., N, Fk = {zi : k < i ≤ N, ji = k}.

k Let us explain brieﬂy what these conditions mean. (i) and (iii) imply that the sequence un ◦ φn 1 bubbles exactly at the points of Fk. Therefore, the condition F1 ⊂ {0} of (i) means that un ◦ φn has at most one bubble point (at 0), that is, the bubble u1 is an endpoint of the tree T .(v) j j means that the bubble u is connected to the bubble u k at the point zk.(iv) means that there are no two bubbles bubbling at the same point. (iii) implies also that there is no loss of energy, j k that is, that the energy of u k at the point zk equals the energy of the bubble u together with all the bubbles accessible from uk.(vii) will be satisﬁed only at the end of the induction, since it says that we have captured all the bubble points at each bubble. Finally, observe that (ii) is the stability condition asserting that if uk is constant, then it is connected to at least three bubbles. Indeed, one point is ∞ by construction and then we have the additional points at Fk.

We think of the un as maps un : C −→ X. We denote by |dun(z)| the norm of the diﬀerential where we use the usual norm in C. We consider in S2 the Fubini-Study metric (identifying S2 with 1 ∼ 2 CP ), which in the open set U1 = C is given by |dun(z)|FS = |dun(z)|(1 + |z| ). Since moreover |du (∞)| is ﬁnite, we must have |du (z)| → 0 when |z| → ∞. Therefore, sup |du (z)| n FS n z∈C n assumes its maximum at some point of C. Therefore, there is a sequence zn ∈ C such that

|dun(zn)| = sup |dun(z)| =: cn z∈C Put v (z) = u (z + z/c ) and observe that sup |dv (z)| = 1 = |dv (0)|. Therefore, some n n n n z∈C n n subsequence converges in the C∞ sense on C = S2 − {∞} to a nonconstant pseudoholomorphic curve v. Passing to a further subsequence we may also assume that E(vn; C − D1) converges. 1 1 1 Then, the functions u (z) = v(1/z) and φ (z) = zn + satisfy (i) − (iii) with F1 = ∅ or n cnz 1 1 F1 = {0}. Conditions (iv) − (vi) are void for N = 1. If F1 = ∅ then un ◦ φn converges to u in ∞ 2 the C sense on all of S and the theorem is proved. So we will assume F1 = {0}.

k k Let l ≥ 1 and suppose by induction that we have constructed u , jk, zk,Fk and {φn}n∈N for k ≤ l in such a way that they satisfy (i) − (vi) but not (vii) with l instead of N. Then, there is a j ≤ l such that Fj 6= {zk : j < k ≤ l, jk = j}. We put

Fj;l := Fj − {zk|j < k ≤ l, jk = j} j Choose any element zl+1 ∈ Fj;l and apply proposition 5.2.1 to the sequence un ◦ φn and the j point z0 = zl+1. Calling the obtained subsequence again by un ◦φn, we get that there exist maps j ψn ∈ G satisfying the conclusion of 5.2.1. Hence, un ◦ φn ◦ ψn converges to a pseudoholomorphic l+1 2 ∞ 2 sphere u : S −→ X in the C sense on S − F , where F ⊂ D1 is a finite set. We also have ∞ 2 that ψn converges to zl+1 in the C sense on compact subsets of C = S − {∞}. Moreover by l+1 j proposition 5.2.3 we have that u (∞) = u (zl+1). This shows that the conditions (i) − (vi) l+1 j with N = l + 1 are satisfied for k ≤ l + 1 with jl+1 := j, Fl+1 = F and φn = φn ◦ ψn. This finishes with the induction.

Now we have to see that the induction terminates after ﬁnitely many steps. Consider at the l-th step of the induction the tree Tl = (j2, ..., jl) and the numbers 78 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

j X m(j; l) = E(u ) + mj(z)

z∈Fj;l

j for j = 1, ..., l where Fj;l is as before. This quantity is the energy of u plus the energy of its bubble points that have not yet been considered. We have m(j; l) = 0 only when uj is constant and all its bubble points have been considered. By (ii) this implies that j is a stable component of Tl (recall that Tl is the tree formed by the vertices v1, ..., vl). Note that by proposition 5.1.2, if m(j; l) 6= 0, then we have m(j; l) ≥ }, where } > 0 is taken as the inﬁmum of the numbers }(Jn) > 0, where we use that }(Jn) converges to }(J) and hence its inﬁmum is greater than 0 after passing to a subsequence of the un. We claim that

l X m(j; l) = E (5.6) j=1

1 If l = 1, we have E = E(u ) + m1(0) = m(1; 1). Suppose that we have proved the identity for some integer l ≥ 1. Note that m(j; l + 1) = m(j; l) whenever j 6= jl+1 and j ≤ l. But

m(jl+1; l + 1) = m(jl+1; l) − mjl+1 (zl+1) and by (iii) with k = l + 1,

m(l + 1; l + 1) = mjl+1 (zl+1) Hence the identity holds for l + 1. Thus we have proved the identity for all l ≥ 1. This means that the total energy at each step is E.

Let us see that the number of edges of Tl is bounded above by 2E/}. Since any bubble that is not constant must have energy at least }, there are at most E/} nonconstant bubbles. Fix a vertex vi of the tree (that is, some bubble). Observe that, since by the stability condition each constant bubble is connected to at least three bubbles, and for each edge of a constant bubble there are (connected to the constant bubble through some path containing that edge) at least two non- constant bubbles. Therefore, there can be at most E/(2}) non-constant bubbles, which means that we have at most 3E/(2}) bubbles. Since at a tree with N vertices we have N − 1 edges, we obtain that we have at most 3E/(2}) − 1 < 2E/} edges. Since at each step of the induction we add an edge to the tree, this shows that the induction terminates after a ﬁnite number of steps.

k k So we have proved the existence of u , jk, zk and {φn}n∈N for k = 1, ..., N such that a suitable subsequence satisﬁes (i) − (vii). We claim now that the subsequence un Gromov converges to the stable map u. First of all we see that u is stable. In order to see this, observe that the vertices k and j with k > j are connected by an edge if and only if j = jk. Moreover, using the notation of the deﬁnition of stable maps, zjkk = zk and zkjk = ∞. So the singular set of the vertex j ∈ {2, ..., N} is Fj ∪ {∞} with Fj as in (vii), while the singular set of the vertex 1 is F1. k But if u is constant, necessarily k ≥ 2 and #Fk ≥ 2. So u is stable.

To finish the proof we must verify the conditions in the definition of Gromov convergence. The (Map) and (Rescaling) conditions follow from (i) and (vi). It remains to verify the (Energy) 0 condition. Denote by Tk the set of vertices k which can be reached from jk by a chain of edges, starting with the one from jk to k. These sets can be defined recursively for k = N,N − 1, ..., 1 by 5.3. GROMOV COMPACTNESS 79

[ Tk := {k} ∪ Ti

ji=n

In particular Tk = {k} whenever there is no i > k such that ji = k. We can prove by induction over the number of elements in Tk that

X k0 mjk (zk) = E(u ) (5.7) 0 k ∈Tk

This follows from (iii) if Zk = ∅ and so Tk = {k}. Assume that we have proved the formula for #Tk ≤ m and suppose #Tk = m + 1. Put {k1, ..., ks} := {i > n : ji = n}.

Then, the set {k1, ..., ks} ∪ {jk} is the set of vertices adjacent to k and #Tkr ≤ m for r = 1, ..., s. Hence, by induction hypothesis,

s k X mk(zk) = E(u ) + mk(zkr ) r=1 for r = 1, ..., s. It follows from (iii) that

s k X mjk (zk) = E(u ) + mk(zkr ) r=1 S Thus, combining the previous formulas and using the fact that Tk = {k} ∪ r Tkr we obtain our claim 5.7. This is just (Energy) for α = jk and β = k. With α = k and β = jk we have Tαβ = {1, ..., N} − Tk and zαβ = ∞ so that (Energy) has the form

k X k0 mk(∞) := lim lim E(un ◦ φn; C − DR) = E(u ) (5.8) R→∞ n 0 k ∈/Tk We can see this as follows

k k X X k0 E − mk(∞) = lim lim E(un ◦ φn; Dr) = E(u ) + mk(z) = mjk (zk) = E(u ) R→∞ n 0 z∈Fk k ∈Tk where we have used the definitions of E and mk(∞) in the first equality, the definition of mk(z) in the second, (iii) in the third and 5.7 in the fourth. Moreover, by 5.6 with l = N we have PN k E = k=1 E(u ), which together with the last identity implies 5.8.

This ﬁnishes the proof of the theorem.

To ﬁnish the chapter, we prove the Gromov compactness theorem for stable maps, which is an easy corollary of the version we have just proved.

Theorem 5.3.2 (Gromov compactness for stable maps). Let (un, zn) be a sequence of Jn-stable maps such that supn E(un) < ∞. Then, (un, zn) has a convergent subsequence. 80 CHAPTER 5. PROOF OF THE GROMOV COMPACTNESS THEOREM

Proof. Put c := supn E(un, zn). Observe that, since the number of edges of the trees are bounded above by 2E (as noted in the proof of the previous theorem), there appear only ﬁnitely many } trees in the stable maps of the sequence, and passing to a subsequence we may assume that all trees are the same. We call it T .

α By the previous theorem, for each α ∈ T the sequence un Gromov converges to a stable map (uα, zα). Now just connect the limit stable maps for diﬀerent edges by adding an additional edge for each pair αEβ. Chapter 6

The non-squeezing theorem

In this section we will give a complete proof (modulo the analytical details about moduli spaces of pseudoholomorphic curves and a result about minimal surfaces) of the celebrated Gromov non-squeezing theorem.

The proof we present in this chapter is essentially the same proof of Gromov, given in [Gro]. The exposition of this chapter closely follows that of [Hum].

6.1 The theorem and its consequences

2m 2m We will denote by Br and Br the open and closed balls of radius r centered at the origin in R2m. We deﬁne also the symplectic cylinders

2m 2 2 2 2 2m−2 2m Zr := {(x1, y1, ..., xn, yn): x1 + y1 < r } = Br × R ⊂ R With these notations we can state the theorem.

Theorem 6.1.1 (Non-squeezing theorem for cylinders). Suppose that there exists a symplectic 2m 2m embedding i : Br −→ ZR . Then, r < R. We will prove a more general result from which this follows.

Theorem 6.1.2 (Non-squeezing theorem). Let M be a closed symplectic manifold of dimension 2m − 2 and assume that π2(M) = 0 (where π2(M) represents the second homotopy group of M). 2m 2 Suppose also that there exists a symplectic embedding i : Br −→ BR × M. Then, r < R. 2m We will see that the non-squeezing theorem for cylinders follows from this version. Since i(Br ) ⊂ 2m 2 R is compact, we can assume that it lies inside BR × C for some cube [a1, b1] × ... × [a2m−2, b2m−2]. Consider the action of the group Γ = {(b1 − a1)n1 + ... + (b2m−2 − 2m−2 2m−2 2m−2 a2m−2)n2m−2} on R by translations. Then, R /Γ ' T , the 2m − 2-dimensional torus. Let us see that this torus inherits a symplectic structure from the symplectic structure in R2m−2. Indeed, put π : R2m−2 −→ T 2m−2 and deﬁne ωT = π∗ω. Since π is a local diﬀeo- morphism and ω is a symplectic form, it follows that ωT is also a symplectic form. Therefore, 2m 2 2m−2 π ◦ i : Br −→ BR × T is also a symplectic embedding (since by choice of the group Γ, π

81 82 CHAPTER 6. THE NON-SQUEEZING THEOREM

2m 2m−2 1 1 is an embedding on i(Br ). Moreover, π2(T ) ' π2(S ) × ... × π2(S ) = 0 (since as it is 1 well-known, πn(S ) = 0 for n > 1). So, by the second version of the non-squeezing theorem, we have that r < R, as wanted.

To realize the importance of the non-squeezing theorem, observe that we have the following

Theorem 6.1.3. Let M be a compact and oriented manifold. Suppose σ and σ0 are two volume forms on M of the same total volume, that is, Z Z σ = σ0 M M Then, there exists a volume and orientation preserving diﬀeomorphism ψ : N −→ N, that is ψ∗σ0 = σ. The proof of this theorem is similar to Darboux’s theorem 1.2.7. Details can be found in [H-Z].

Observe that, since a symplectic form determines a volume form in a symplectic manifold, any symplectomorphism between compact symplectic manifolds must be volume preserving. Tak- ing this into account, we see that the non-squeezing theorem gives us a fundamental diﬀerence between symplectomorphisms and volume preserving diﬀeomorphisms, because any ball can be embedded in a volume preserving way into a cylinder, regardless of their radii. This gives a hint that there are non-trivial symplectic invariants, that are in some sense 2-dimensional.

We now deﬁne these non-trivial symplectic invariants, the symplectic capacities. Deﬁnition 6.1.4. A symplectic capacity c is a function that assigns to each symplectic manifold (X, ω) a number in [0, ∞] satisfying 1. If there is a symplectic embedding i :(X, ω) −→ (X0, ω0) and dim X = dim X0 then c(X, ω) ≤ c(X0, ω0). 2. c(X, λω) = c(X, ω)

2n 2n 3. c(B (1), ω0) > 0 and c(Z (1), ω0) < ∞. Observe that the first axiom implies that two symplectomorphic manifolds have the same symplectic capacity, hence it is a real symplectic invariant. Observe also that the volume cannot be 2n a symplectic capacity, since the third axiom implies that (Z (1), ω0) has finite capacity, while it has infinite volume. Symplectic capacities must be thought of as measuring a two-dimensional quantities.

It is not trivial to show that there exist symplectic capacities. However, with the non-squeezing theorem, we can prove that they exist. Deﬁnition 6.1.5. The Gromov width of a symplectic manifold (X, ω) is deﬁned by

2 2n wG(X, ω) = wG(X) = sup{πr : B (r) embeds symplectically in X}

Proposition 6.1.6. The Gromov width is a symplectic capacity. Proof. Indeed, the Gromov width of a symplectic manifold is a symplectic capacity with the desired additional condition. That it satisfies the first and second axioms is immediate from the definition. The fact that it satisfies the third axiom follows from the non-squeezing theorem. 6.2. PROOF OF THE THEOREM 83

6.2 Proof of the theorem

In this section we will give the proof of the non-squeezing theorem. Let (M, ω) be a closed symplectic manifold with π2(M) = 0. Consider the symplectic product

(V, Ω) := (S2 × M, σ ⊕ ω) where σ is any symplectic structure on S2.

2 We denote by A ∈ H2(V ) the homology class represented by the inclusion iS2 : S −→ V deﬁned by iS2 (z) = (z, q0).

The main ingredient is the following result telling us that there exist a lot of pseudoholomorphic curves.

Theorem 6.2.1. For each Ω-compatible almost complex structure JV on V and any given point 2 p0 ∈ V there exists an JV -holomorphic curve u : S −→ V in the homology class A containing p0 in its image.

Recall that P : M(A, J ) −→ J is the canonical projection. Note that if JM is an almost complex 2 2 structure on M, then, putting σq : S −→ S × M deﬁned by σq(z) = (σ(z), q) for σ ∈ G, we −1 have that P (j ⊕ JM ) = {σq : σ ∈ G, q ∈ M}. In order to prove this, observe that if u is such 2 that P (u) = j ⊕ JM (so that u is a j ⊕ JM -holomorphic map) we can consider π ◦ u : S −→ M 2 where π : S × M −→ M is the canonical projection, which is an JM -holomorphic curve. Since π2(M) = 0, we have that π ◦ u is homotopic to a constant map. Since pseudoholomorphic maps minimize the energy in its homology class, it follows that π ◦ u must be constant. On the other hand, π0 ◦ u : S2 −→ S2, where π0 : S2 × M −→ S2 is the canonical projection is a j-holomorphic map, it follows that π0 ◦ u =: σ ∈ G, as wanted.

We will use the following fact without proof.

Lemma 6.2.2. Suppose JM is integrable on an open set U0 ⊂ M. Then, for each σ ∈ G and each q ∈ U0, the j ⊕ JM -holomorphic curve σq is a regular point of P . The next theorem, which is of crucial importance in our proof, is a corollary of the Gromov compactness theorem. In particular, it tells us that the spaces M(A, J) × S2/G are already compact, so there is no need to consider stable curves.

2 Proposition 6.2.3. Suppose that {[un, zn]}n≥1 is a sequence in M(A, J )×S /G and that some subsequence {P (un)}n≥1 converges in J . Then, {[un, zn]}n≥1 has a convergent subsequence.

Proof. Put J = limn P (un). Under the assumption of the theorem we have then that un is a sequence of Jn-holomorphic maps, where Jn converge to J.

Since each un represents A and Jn is Ω-compatible, we have that the energy of un is given by < Ω, A >, so the sequence has bounded energy. By the compactness theorem, we have that un Gromov converges to a stable curve u of energy E(u) =< Ω, A >. Let us see that the stable curve u is actually a pseudoholomorphic sphere. Indeed, suppose it has at least two non-constant components, u1 and u2. Then we have 84 CHAPTER 6. THE NON-SQUEEZING THEOREM

0 << Ω, [u1] >= E(u1) < E(u) =< [Ω], A > Therefore there is a homology class with energy positive but strictly smaller than that of A. 2 This leads to a contradiction as follows. Since π2(M) = 0, we have that π2(V ) = π2(S × M) ' 2 π2(S ) ' Z, where the last isomorphism is via the degree of the map. Suppose now that g : S2 −→ V is a pseudoholomorphic curve and that π ◦ g : S2 −→ S2 is a map of de- 2 gree n. Then, g∗[S ] = (deg(g))A (since A is a generator of H2(V )), which implies that 2 E(g) =< [Ω], g∗[S ] >= (deg(g)) < [Ω], A > cannot lie strictly between 0 and < [Ω], A >.

Hence, u is actually a J-holomorphic sphere. The assertion that un Gromov converges to u means that there exist Moebius transformations ψn ∈ G such that un ◦ ψn converge to u in the ∞ 2 −1 C sense. Since we are in M(A, J ) × S /G, we have that [un, zn] = [un ◦ ψn, ψn (zn)]. By −1 2 passing to a subsequence if necessary, we may assume that ψn (zn) converges to some z ∈ S 2 (by compacity of S ). Therefore, we have that [un, zn] converge to [u, z], as wanted. Using this proposition, we will give a proof of theorem 6.2.1. Before we go into the details, let us give a brief outline of the proof. First of all, note that if the almost complex structure JV is 2 of the form j ⊕ JM , where j is the usual almost complex structure on S and JM is a regular almost complex structure in M, then the theorem is trivial. This is because, in this case, putting 2 2 p0 = (r0, q0) it suﬃces to consider a pseudoholomorphic map u : S −→ V = S × M of the form u(z) = (σ(z), q0) for some σ ∈ G. Now, for a general almost complex structure JV we will show by using a cobordism argument with the almost complex structure j ⊕ JM that there are Jb-holomorphic curves passing through pb for Jb regular arbitrarily close to JV and pb arbitrarily close to p0. Then, choosing a sequence of such maps and by applying proposition 6.2.3, we will show that we have a convergent subsequence and the limit pseudoholomorphic map has the desired properties.

Let us carry out this program in detail. First of all, observe that if JM is an almost complex structure on M and j is the usual almost complex structure in S2, we have

−1 P (j ⊕ JM ) = {σq : σ ∈ G, q ∈ M} 2 where σq : S −→ V is deﬁned by σq(z) = (σ(z), q). It is clear that every σq is a j ⊕ JM - 2 holomorphic map. Conversely, if u is a j ⊕ JM -holomorphic map, consider πM ◦ u : S −→ M 2 where πM : S × M −→ M is the canonical projection. Then, πM ◦ f is an JM -holomorphic map in M. But since π2(M) = 0, πM ◦ f is homotopic to a constant map. It follows that πM ◦ u has 2 2 the zero energy, hence it is constant. On the other hand, πS2 ◦ u : S −→ S is a j-holomorphic map, hence a Moebius transfomation, as wanted.

Lemma 6.2.4. Given neighbourhoods U of j ⊕ JM in J and U of p0 in V , there exist an almost complex structure Ja ∈ U ∩ Jreg and a point pa ∈ U such that pa is a regular value of evJa and ev−1(p ) has exactly one point. Ja a Proof. Consider the following map

Φ: M(A, J ) × S2 −→ J × V deﬁned by Φ(f, z) = (P (f), f(z)). Since Φ is obtained from P and the evaluation map, we see −1 that it is a Fredholm map. From the previous description of P (j ⊕ JM ) and lemma 6.2.2, we can see that (j ⊕ Im, p0) is a regular value of Φ. Therefore, there is an open connected 6.2. PROOF OF THE THEOREM 85

neighbourhood V of (idq0 , z0) = (j ⊕ JM , p0) in J × V such that Φ|V : V −→ W is a submersion. Then, by a form of the inverse function theorems for Banach manifolds, there is a diﬀeomorphism ξ : W × B −→ V. By shrinking the neighbourhoods, we may assume without loss of generality −1 n that B = Φ|V (j ⊕ JM , p0) is a connected open ball in some R such that

Φ ◦ ξ : W × B −→ W is the projection onto W. Observe that dim B = dim G, because there is only one j⊕JM -holomorphic curve passing through p0 modulo the action of G. Choose (w, b) ∈ W ×B. Then the set of points in {w}×B equivalent to (w, b) by the action of G form an open set in {w} × B because dim B = dim G, and moreover it is closed, so by connectedness, it is all of {w} × B. Therefore, given (J, p) ∈ W, we have that there is a unique J-holomorphic curve in V passing through p modulo G.

Now the Sard-Smale theorem tells us that there exists a sequence (Jn) in Jreg converging to 2 j ⊕ JM and elements (un, zn) ∈ M(A, Jn) × S converging to (idq0 , z0) such that [un, zn] are regular points of evJn .

Suppose now that the lemma is false, that is, that there exist neighbourhoods of U and of p0 such that for any almost complex structure J and point p in the corresponding neighbourhoods, p is not a regular value of evJ . Then, from some n on, we have that there are elements 0 0 2 0 0 0 0 (un, zn) ∈ M(A, In) × S with evJn ([un, zn]) = evJn ([un, zn]) and [un, zn] a singular point of 2 0 0 evJn . Using proposition 6.2.3 and the compactness of S , we may assume that (un, zn) converge 0 0 2 0 0 0 0 to some (u, , z ) ∈ M(A, | ⊕ JM) × S . We have that u (z ) = limn un(zn) = limn un(zn) = p0 and using that there is only one j ⊕ JM -holomorphic curve through p0, we may assume (after a 0 0 0 0 reparametrization) that (u , z ) = (idq0 , z0). Hence, (un, zn) ∈ V for n large enough, and then 0 0 0 0 (un, zn) = (un, zn), which is a contradiction since [un, zn] is singular for evJn while [un, zn] is regular. This ﬁnishes the proof of the ﬁrst assertion.

The second assertion is now trivial, since we have already seen that there is a unique Ja- holomorphic curve passing through pa, modulo reparametrization by G.

Recall that the set of regular almost complex structures is open in J , so we can pick a regular almost complex structure Jb in any neighbourhood of JV . By the Sard-Smale theorem, the set of regular values for evJb is also dense, hence we can pick such a regular value pb in any neighbourhood of p0. Now we have

Lemma 6.2.5. Let Jb and pb as before. Then there exists an Jb-holomorphic curve representing A and passing through pb.

Proof. We choose pa and Ja in a neighbourhood of (p0, j⊕JM ) satisfying the previous lemma. By the path-connectedness of J and the transversality theorem, there is a smooth path α : [0, 1] −→ −1 J joining Ja to Jb transverse to the P . Then, M(A, α) := P (α([0, 1])) is a ﬁnite-dimensional smooth manifold with boundary, and its boundary is ∂M(A, α) = M(A, Ja) ∪ M(A, Jb), where the union is disjoint. Observe that G acts in M(A, α)×S2 as always, by g·(u, z) = (u◦g−1, g(z)) , and this action gives rise to a manifold M(A, α) × S2/G. We can consider again an evaluation map

2 evα : M(A, α) × S /G −→ V × [0, 1] 86 CHAPTER 6. THE NON-SQUEEZING THEOREM

by evα([u, z]) = (u(z), t) is f is an α(t)-holomorphic curve. This is well-deﬁned since α is transversal to P . Then, the points (pa, 0) and (pb, 1) are regular values of evα, since pa and 2 pb are regular values of Ja and Jb respectively. Since V × S is connected, we can choose a path β : [0, 1] −→ V × [0, 1] from pa to pb transverse to evα and transverse to the boundary V × {0} ∪ V × {1}.

−1 Put now N := evα (β([0, 1])). Then, N is a smooth submanifold of M(A, α) and has boundary 2 2 ∂N = N ∩ ∂M(A, α) = (N ∩ (M(A, Ja) × S )/G) ∪ (N ∩ (M(A, Jb) × S )/G). Observe that by applying proposition 6.2.3, we obtain that any subsequence of M(A, α) contains a convergent subsequence, so M(A, α) is compact. Then N, being a closed submanifold of M(A, α) is compact too. By the previous lemma, the component N ∩(M(A, I )×S2)/G) = ev−1(p ) of the boundary a Ja a of N consists on exactly one point. Therefore, N must be one-dimensional and its boundary is 2 a disjoint union of an even number of points. This implies that (N ∩ (M(A, Jb) × S )/G 6= ∅. That is, there is an Jb-holomorphic curve passing through pb, as wanted.

Now we can easily ﬁnish the proof of theorem 6.2.1.

Proof of theorem 6.2.1. The previous lemma implies that there is a sequence (Jn, pn) in J × V converging to (JV , p0) and a sequence un of Jn-holomorphic curves representing the homology class A passing through pn. Let zn such that un(zn) = pn. Use proposition 6.2.3 to extract a subsequence of the (un, zn) such that there exists ψn ∈ G satisfying that un ◦ ψn converges to ∞ −1 2 a JV -holomorphic curve u in the C sense and ψ (zn) converges to a point z ∈ S . Then, −1 u(z) = limn(un ◦ψn)(ψn (zn)) = limn pn = p0. So u is the wanted pseudoholomorphic curve.

Now that we know that there exist plenty of pseudoholomorphic curves, we are ready to prove theorem 6.1.2. We will use the following result from the theory of minimal surfaces, which we quote without proof. Recall that a surface in R2n is proper if each closed ball in R2n contains a compact portion of the surface with respect to its intrinsic topology.

Theorem 6.2.6 (Monotonicity theorem). Any proper minimal surface passing through the origin in B2n(R) ⊂ Cn has its area greater or equal than πR2. For a proof, see [M-P], Theorem 2.6.2. Observe that for a minimal surface S ⊂ R2n, its area is given by Z A(S) = ω0 S 2n where ω0 is the restriction to S of the standard symplectic form of R .

Suppose we have a holomorphic map u : S −→ Cn. Then, we have: Z Z ∗ A(S) = ω0 = u ω0 = E(u) S S This tells us that the area of a surface which is the domain of a holomorphic curve equals the energy of the curve. Moreover, since holomorphic curves (which are just J0-holomorphic curves) are energy minimizing in their homology class, this means that S is an area minimizing surface, that is, a minimal surface. We will use this identity in the following proof. 6.2. PROOF OF THE THEOREM 87

Proof of theorem 6.1.2. Assume that there is a symplectic embedding i : B2m(r) −→ B2(R)×M. Choose > 0 and a symplectic structure σ on S2 such that A(S2, σ) = πR2 + . Note that this is possible simply by rescaling the symplectic form (which in the case of a surface is the same as an area form). Hence, we have A(B2(R)) = πR2 < A(S2, σ), so we can embed symplectically (which in this case is just the same as volume-preserving) B2(R) into S2. In this way, we will view B2(R) as a subset of (S2, σ). So composing the two embeddings, we have an embedding (still denoted by i) i : B2m(r) −→ (V, Ω) = (S2 × M, σ ⊕ ω).

We will choose now an Ω-compatible almost complex structure J on V in such a way that ∗ 2m J0|B2m(r−) = i (J|i(B2m(r−)). Let us see how to do this. Pick, for each point x ∈ i(B (r − )), a coordinate chart (Ux, ξx) and a partition of unity {ρx}x subordinated to the {Ux}x. Deﬁne a ∗ 2m 2m metric gx in each of the Ux with the condition that i (gx|Ux∩i(B (r)) = g0|ξx(Ux)∩B (r) and put P ∗ g = x ρxgx. Then, we have obtained a metric g on V such that i (g|B2m(r−))) = g0|B2m(r−). Now apply the construction in the proof of proposition 1.3.3 to obtain an Ω-compatible almost complex structure J from g. Since starting from ω0 and g0 one obtains the usual almost complex structure J0, we see that our J satisfy the desired conditions.

Observe now that i(B2m(r − )) is holomorphically isometric to the Euclidean ball B2m(r − ), 2m where we are considering in i(B (r −)) the metric gJ induced by the almost complex structure J. Indeed, the map i is an isometry because if u, v ∈ TB2m(r − ) we have

∗ i (gJ (u, v)) = gJ (di(u), di(v)) = Ω(di(u), Jdi(v))

= Ω(di(u), di(J0(v))) = ω0(u, J0(v)) = g0(u, v) where we have used that i is symplectic and that Jdi = diJ0. The holomorphicity follows from ∗ 2m the fact that i (I) = J0, where J0 is the usual almost complex structure in R .

Now we apply theorem 6.2.1 to obtain a J-holomorphic curve u : S2 −→ V representing the ho- 2m mology class A and passing through i(0). Observe that, since i(B (r − )) is a 2m-dimensional 2m submanifold with boundary of V , S := u−1(i(B (r−))) is a proper 2-dimensional submanifold with boundary of S2 (it cannot be the whole S2, in other words, the image of u is not completely 2m contained in i(B (r)), because if this were the case u would be constant, being a pseudoholo- 2m morphic curve homotopic to a constant map). Now consider u|S : S −→ B (r − ), where we 2m 2m have identiﬁed i(B (r−)) with the euclidean ball B (r−) via the holomorphic isometry i−1. 2m Then, u|S is a J0-holomorphic curve, and since J0 is the usual almost complex structure in R , we have that u|S is in fact a holomorphic curve. Moreover, since holomorphic maps minimize energy in their homology class, and in the case of holomorphic maps, energy coincides with the 2m area, we have that u|S is area minimizing among all the smooth maps φ : S −→ B (r − ) which satisfy φ|∂S = u|∂S. This means that S is a minimal surface.

Therefore, we can apply the monotonicity lemma for minimal surfaces, to conclude that

2 π(r − ) ≤ A(S) = E(u|S) From here, we get the following chain of inequalities: Z 2 ∗ 2 π(r − ) ≤ E(u|S) < E(u) =< [Ω], A >= incl Ω = πR + . S2 88 CHAPTER 6. THE NON-SQUEEZING THEOREM

Therefore, since this is valid for any , it follows that r ≤ R, as claimed. Bibliography

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